On the measure-preserving mappings with odd dimension

In this paper, using two methods: LCN'S (Lyapunov characteristic numbers) method and slice cutting method, we study numerically two mappings with odd dimension: $$T_1 :\left\{ {\begin{array}{*{20}c} {x_{n + 1} = x_n + z_n ,} \\ {y_{n + 1} = y_n + x_{n + 1} , (\bmod 2\pi )} \\ {z_{n + 1} = z_n + A\sin y_{n + 1} ,} \\ \end{array} } \right. T_2 :\left\{ {\begin{array}{*{20}c} {x_{n + 1} = x_n + y_n + B \sin z_n ,} \\ {y_{n + 1} = y_n + A \sin x_{n + 1} , (\bmod 2\pi ),} \\ {z_{n + 1} = z_n + B \sin y_{n + 1} ,} \\ \end{array} } \right.$$ whereA, B are parameters. For the mappingT ₁ the whole region is stochastic; however, we find two-dimensional invariant manifolds for the mappingT ₂.     

A note on first-order normalizations of perturbed Keplerian systems

Techniques are developed to facilitate the transformation of a perturbed Keplerian system into Deláunay normal form at first order. The implicit dependence of the Hamiltonian on 1, the mean anomaly, through the explicit variable f, the true anomaly, or E, the eccentric anomaly, is removed through first order for terms of the form:

Scanner observations of the classical cepheids RT Aur and T Vul

The effective temperatures of the classical Cepheids RT Aur and T Vul have been determined by a comparison of their spectral scans with appropriate model atmospheres. The radii of the stars have been determined through the Wesselink method. Using these temperatures and the Wesselink radii, the luminosities of the stars have been determined. These radii estimates, including the radii of SU Cas (Joshi & Rautela 1980) andζ Gem (unpublished) fit better in the theoretical period-radius relationship given by Cogan (1978), as compared to earlier determinations of Wesselink radii. The pulsation masses and evolutionary masses of the stars have been calculated. The pulsation to evolutionary mass ratio is derived to be 0.85. Based on the effective temperatures obtained by us at different phases of the stars aθ _c ? (B-V)₀ relationship is found of the form, \(\begin{gathered} \theta _e = 0.274 (B - V)_0 + 0.637 \\ \pm 0.011 \pm 0.007 \\ \end{gathered} \)      

A perturbed extension of hyperbolic twist mappings

In this paper we discuss a perturbed extension of hyperbolic twist mappings to a 3-dimensional measure-preserving mapping $$\begin{array}{*{20}c} {T:\left\{ {\begin{array}{*{20}c} {x_{n + 1} = s(x_n \cos \varphi _n - y_n \sin \varphi _n ) + A\cos z_n ,} \\ {y_{n + 1} = s^{ - 1} (x_n \sin \varphi _n + y_n \cos \varphi _n ) + B\sin z_n ,} \\ {z_{n + 1} = z_n + C\cos (x_{n + 1} + y_{n + 1} ) + D,(\bmod 2\pi )} \\ \end{array} } \right.} \\ {\varphi _n = (x_n^2 + y_n^2 )^k } \\ \end{array}$$ wheres, k are parameters andA, B, C, D are perturbation parameters. We find that the ordered regions near the fixed point of the hyperbolic twist mapping is destroyed by the perturbed extension more easily than the ones distant from it. The size of the ordered region decreases with increasing perturbation parameters and is insensitive to the parameterD for the same parametersA, B, C.     

Attractors in a dissipative dynamical system with three dimensions

We study an extension of the Hénon mapping to a dissipative dynamical system with three-dimensions and discuss the behavior of the attractors of the Hénon mapping in the extended mapping $$T:\left\{ {\begin{array}{*{20}c} {X_{i + 1} = Y_i + 1 - AX_i^2 + C cos Z_i } \\ {Y_{i + 1} = BX_i + D \sin Z_i } \\ {Z_{i + 1} = Z_i + E \sin Y_{i + 1} + F, (\bmod 2\pi ).} \\ \end{array} } \right.$$ The results show that the strange attractor is destroyed by perturbed extension more easily than the trivial attractor and the invariant manifold of the conservative dynamical system.     

Global solution of the ideal resonance problem

If a dynamical problem ofN degress of freedom is reduced to the Ideal Resonance Problem, the Hamiltonian takes the form 1 $$\begin{array}{*{20}c} {F = B(y) + 2\mu ^2 A(y)\sin ^2 x_1 ,} & {\mu \ll 1.} \\ \end{array} $$ Herey is the momentum-vectory _k withk=1,2?N, x ₁ is thecritical argument, andx _k fork>1 are theignorable co-ordinates, which have been eliminated from the Hamiltonian. The purpose of this Note is to summarize the first-order solution of the problem defined by (1) as described in a sequence of five recent papers by the author. A basic is the resonance parameter α, defined by 1 $$\alpha \equiv - B'/\left| {4AB''} \right|^{1/2} \mu .$$ The solution isglobal in the sense that it is valid for all values of α² in the range 1 $$0 \leqslant \alpha ^2 \leqslant \infty ,$$ which embrances thelibration and thecirculation regimes of the co-ordinatex ₁, associated with α² < 1 and α² > 1, respectively. The solution includes asymptotically the limit α² → ∞, which corresponds to theclassical solution of the problem, expanded in powers of ε ≡ μ², and carrying α as a divisor. The classical singularity at α=0, corresponding to an exact commensurability of two frequencies of the motion, has been removed from the global solution by means of the Bohlin expansion in powers of μ = ε^1/2. The singularities that commonly arise within the libration region α² < 1 and on the separatrix α² = 1 of the phase-plane have been suppressed by means of aregularizing function 1 $$\begin{array}{*{20}c} {\phi \equiv \tfrac{1}{2}(1 + \operatorname{sgn} z)\exp ( - z^{ - 3} ),} & {z \equiv \alpha ^2 } \\ \end{array} - 1,$$ introduced into the new Hamiltonian. The global solution is subject to thenormality condition, which boundsAB″ away from zero indeep resonance, α² < 1/μ, where the classical solution fails, and which boundsB′ away from zero inshallow resonance, α² > 1/μ, where the classical solution is valid. Thedemarcation point 1 $$\alpha _ * ^2 \equiv {1 \mathord{\left/ {\vphantom {1 \mu }} \right. \kern-\nulldelimiterspace} \mu }$$ conventionally separates the deep and the shallow resonance regions. The solution appears in parametric form 1 $$\begin{array}{*{20}c} {x_\kappa = x_\kappa (u)} \\ {y_1 = y_1 (u)} \\ {\begin{array}{*{20}c} {y_\kappa = conts,} & {k > 1,} \\ \end{array} } \\ {u = u(t).} \\ \end{array} $$ It involves the standard elliptic integralsu andE((u) of the first and the second kinds, respectively, the Jacobian elliptic functionssn, cn, dn, am, and the Zeta functionZ (u).     

A few observations of and remarks on V1500 Cygni

The development of the post-nova light curve of V1500 Cyg inUBV andHβ, for 15 nights in September and October 1975 are presented. We confirm previous reports that superimposed on the steady decline of the light curve are small amplitude cyclic variations. The times of maxima and minima are determined. These together with other published values yield the following ephemerides from JD 2 442 661 to JD 2 442 674: $$\begin{gathered} {\text{From}} 17 {\text{points:}} {\text{JD}}_{ \odot \min } = 2 442 661.4881 + 0_{^. }^{\text{d}} 140 91{\text{n}} \hfill \\ \pm 0.0027 \pm 0.000 05 \hfill \\ {\text{From}} 15 {\text{points:}} {\text{JD}}_{ \odot \max } = 2 442 661.5480 + 0_{^. }^{\text{d}} 140 89{\text{n}} \hfill \\ \pm 0.0046 \pm 0.0001 \hfill \\ \end{gathered} $$ with standard errors of the fits of ±0 _. ^d 0052 for the minima and ±0 _. ^d 0091 for the maxima. Assuming V1500 Cyg is similar to novae in M31, we foundr=750 pc and a pre-nova absolute photographic magnitude greater than 9.68.     

On equivalent widths in a penumbral spectrum (a test of the Moe-Maltby model)

From new observational material we made a curve of growth analysis of the penumbra of a large, stable sunspot. The analysis was done relative to the undisturbed photosphere and gave the following results (⊙ denotes photosphere, * denotes penumbra): $$\begin{gathered} (\theta ^ * - \theta ^ \odot )_{exe} = 0.051 \pm 0.007 \hfill \\ {{\xi _t ^ * } \mathord{\left/ {\vphantom {{\xi _t ^ * } {\xi _t }}} \right. \kern-\nulldelimiterspace} {\xi _t }}^ \odot = 1.3 \pm 0.1 \hfill \\ {{P_e ^ * } \mathord{\left/ {\vphantom {{P_e ^ * } {P_e ^ \odot = 0.6 \pm 0.1}}} \right. \kern-\nulldelimiterspace} {P_e ^ \odot = 0.6 \pm 0.1}} \hfill \\ {{P_g ^ * } \mathord{\left/ {\vphantom {{P_g ^ * } {P_g }}} \right. \kern-\nulldelimiterspace} {P_g }}^ \odot = 1.0 \pm 0.2 \hfill \\ \end{gathered} $$ The results of the analysis are in satisfactory agreement with the penumbral model as published by Kjeldseth Moe and Maltby (1969). Additionally we tested this model by computing the equivalent widths of 28 well selected lines and comparing them with our observations.     

Some further reduction formulas for certain classes of-generalized multiple hyper geometric series arising in physical,astrophysical, and quantum chemical applications

The multivariable hypergeometric function $$F_{q_0 :q_1 ;...;q_n }^{P_0 :P_1 ;...;P_n } \left( {\begin{array}{*{20}c} {x_1 } \\ \vdots \\ {x_n } \\ \end{array} } \right),$$ considered recently by A. W. Niukkanen and H.M. Srivastava, is known to provide an interesting unification of the generalized hypergeometric function_p F _q of one variable, Appell and Kampé de Fériet functions of two variables, and Lauricella functions ofn variables, as also of many other hypergeometric series which arise naturally in various physical, astrophysical, and quantum chemical applications. Indeed, as already pointed out by Srivastava, this multivariable hypergeometric function is an obvious special case of the generalized Lauricella function ofn variables, which was first introduced and studied by Srivastava and M. C. Daoust. By employing such fruitful connections of this multivariable hypergeometric function with much more general multiple hypergeometric functions studied in the literature rather systematically and widely, Srivastava presented several interesting and useful properties of this function, most of which did not appear in the work of Niukkanen. The object of this sequel to Srivastava's work is to derive a further reduction formula for the multivariable hypergeometric function from substantially more general identities involving multiple series with essentially arbitrary terms. Some interesting connections of the results considered here with those given in the literature, and some indication of their applicability, are also provided.     

Balloon observations of Sco X-1 in the energy interval 17–106 keV

The paper presents the intensity and spectral nature of the X-ray emission from Sco X-1 in the energy interval 17–106 keV based on the observations made by a balloon borne scintillation telescope system flown on November 15, 1971 from Hyderabad, India. In the 25–53 keV interval, the spectral distribution is observed to correspond to akT value of keV assuming the shape to be exponential. Over the complete energy range of observation, a power law function with the value of exponent equal to 3.6±0.5 seems to yield an adequate fit. Comparing the present data with those obtained elsewhere, the temporal characteristics of the X-ray emission from Sco X-1 are discussed.     

Orbit and physical characteristics of the components of the massive Algol V622 Per,a member of the open star cluster χ Per

Analysis of the radial velocities based on spectra of high (near the H _α line) and moderate (4420–4960 Å) resolutions supplemented by the published radial velocities has revealed the binarity of a bright member of the young open star cluster χ Per, the star V622 Per. The derived orbital elements of the binary show that the lines of both components are seen in its spectrum, the orbital period is 5.2 days, and the binary is in the phase of active mass exchange. The photometric variability of the star is caused by the ellipsoidal shape of its components. Analysis of the spectroscopic and photometric variabilities has allowed the absolute parameters of the binary’s orbit and its components to be found. V622 Per is shown to be a classical Algol with moderate mass exchange in the binary. Mass transfer occurs from the less massive (\({M_1} = 9.1 \pm 2.7{M_ \odot }\)) but brighter (\(\log {L_1} = 4.52 \pm 0.10{L_ \odot }\)) component onto the more massive (\({M_2} = 13.0 \pm 3.5{M_ \odot }\)) and less bright (\(\log {L_2} = 3.96 \pm 0.10{L_ \odot }\)) component. Analysis of the spectra has confirmed an appreciable overabundance of CNO-cycle products in the atmosphere of the primary component. Comparison of the positions of the binary’s components on the T _eff–log g diagram with the age of the cluster χ Per points to a possible delay in the evolution of the primary component due to mass loss by no more than 1–2Myr.     

Spot modelling and elements of the RS CVn eclipsing binary WY Cancri

Results of analysis of photoelectric observations of the RS CVn eclipsing binary WY Cancri in the standard passbands ofUBV during 1973-74, 1976-79 and inUBVRI during 1984-86 are reported. A preliminary analysis of the eclipses suggested the primary eclipse to be transit. A study of the percentage contribution of the distortion wave amplitudes in all the colours with respect to the luminosities of both components, showed the hotter component to be the source of the distortion wave. The clean (wave removed) light curves of different epochs have not merged, suggesting residual effects of spot activity. The reason for this is attributed to the presence of either (1) polar spots or (2) small spots uniformly distributed all over the surface of the hotter component. This additional variation is found to have a periodicity of about 50 years or more. The distortion waves in yellow colour are modelled according to Budding’s (1977) method. For getting the best fit of the observations and theory, it was found necessary to assume three or four spots on the surface of the hot component. Out of these four spot groups, three are found to have direct motion with migration periods of 1.01, 1.01 and 2.51 years while the fourth one has a retrograde motion with a migration period of 3.01 years. From these periods and the latitudes of the spots derived from the model a co-rotating latitude of 4ℴ is obtained. The temperatures of these spots are found to be lower than that of the photosphere by about 700ℴK to 800ℴK. Assuming the light curve of 1985-86, which is the brightest of all the observed seasons, to be least affected by the spots, the light curves of the other seasons are all brought up to the quadrature level of this season by applying suitable corrections. The merged curves in theUBVRI colours are analysed for the elements by the Wilson-Devinney method. This analysis yielded the following absolute elements:

Formation of Thorne–Żytkow objects in close binaries

Thorne–?ytkow objects (T?Os), originally proposed by Thorne and ?ytkow, may form as a result of unstable mass transfer in a massive X-ray binary after a neutron star (NS) is engulfed in the envelope of its companion star. Using a rapid binary evolution program and the Monte Carlo method, we simulated the formation of T?Os in close binary stars. The Galactic birth rate of T?Os is about \(1.5\times 10^{-4}~\hbox {yr}^{-1}\). Their progenitors may be composed of a NS and a main-sequence star, a star in the Hertzsprung gap or a core-helium burning, or a naked helium star. The birth rates of T?Os via the above different progenitors are \(1.7\times 10^{-5}\), \(1.2\times 10^{-4}\), \(0.7\times 10^{-5}\), \(0.6\times 10^{-5}~\hbox {yr}^{-1}\), respectively. These progenitors may be massive X-ray binaries. We found that the observational properties of three massive X-ray binaries (SMC X-1, Cen X-3 and LMC X-4) in which the companions of NSs may fill their Roche robes were consistent with those of their progenitors.     

Effect of oblateness on the non-linear stability of L 4 in the restricted three-body problem

Non-linear stability of the libration point L ₄ of the restricted three-body problem is studied when the more massive primary is an oblate spheroid with its equatorial plane coincident with the plane of motion, Moser's conditions are utilised in this study by employing the iterative scheme of Henrard for transforming the Hamiltonian to the Birkhoff's normal form with the help of double D'Alembert's series. It is found that L ₄ is stable for all mass ratios in the range of linear stability except for the three mass ratios: $$\begin{gathered} \mu _{c1} = 0.0242{\text{ }}...{\text{ }}{}^{{\text{\_\_}}}0.1790{\text{ }}...{\text{ }}A_1 , \hfill \\ \mu _{c2} = 0.0135{\text{ }}...{\text{ }}{}^{{\text{\_\_}}}0.0993{\text{ }}...{\text{ }}A_1 , \hfill \\ \mu _{c3} = 0.0109{\text{ }}...{\text{ }}{}^{{\text{\_\_}}}0.0294{\text{ }}...{\text{ }}A_1 . \hfill \\ \end{gathered} $$      

Gamma-ray burst measurements with A 6.3 m2 array

Two flights from Alice Springs, Australia, were achieved in November 1977 and November 1978 with a plastic scintillator -burst detector, effective area 6.3 m², thickness 5 cm, energy response in the range 50 keV to 2 MeV. In 33 hr of good, high altitude data, two bursts were detected, yielding a rate corrected to an isotropic flux of at a size of 8.5×10^–9 erg cm^–2. One event, seen at 22.14 on 15 Nov 1978, was confirmed by spacecraft measurements. The second, too small to be detected by spacecraft, arrived from 0 hr RA, –13.2° Decl. ±12° and possibly comes from a confirmed -burst source location. A galactic origin with a source distribution originating from a relatively thick disk, is favoured by these results.     

Supernovae in close binaries

The impact of a supernova shell onto 2.82M _⊙ and 20.0M _⊙ main-sequence stars is investigated for various initial orbital separations, and various supernova shell masses and velocities. The inelastic collision between the star and the supernova shell, the shock propagation into the companion star, and other forms of momentum transfer such as the rocket effect are considered. The total momentum transfer due to the supernova is insufficient to eject the companion from the binary as long as the companion retains most of its mass, regardless of the initial orbital separation. Ejection of the companion may occur if the companion is nearly destroyed. Even in contact binaries destruction does not necessarily occur, and if the orbital separation exceeds 10¹² cm, destruction of the companion becomes quite unlikely.     

Apsidal motion in the close binary IT cassiopeiae

In 1982 and 1993, we carried out highly accurate photoelectric WBVR measurements for the close binary IT Cas. Based on these measurements and on the observations of other authors, we determined the apsidal motion $\left[ {\dot \omega _{obs} = {{(11\mathop .\limits^ \circ 0 \pm 2\mathop .\limits^ \circ 5)} \mathord{\left/ {\vphantom {{(11\mathop .\limits^ \circ 0 \pm 2\mathop .\limits^ \circ 5)} {100 years}}} \right. \kern-0em} {100 years}}} \right]$ . This value is in agreement with the theoretically calculated apsidal motion for these stars $\left[ {\dot \omega _{th} = {{(14^\circ \pm 3^\circ )} \mathord{\left/ {\vphantom {{(14^\circ \pm 3^\circ )} {100 years}}} \right. \kern-0em} {100 years}}} \right]$ .     

The Non-Linear Stability of L4 in the Restricted Three-Body Problem when the Bigger Primary is a Triaxial Rigid Body

The non-linear stability of L ₄ in the restricted three-body problem has been studied when the bigger primary is a triaxial rigid body with its equatorial plane coincident with the plane of motion. It is found that L ₄ is stable in the range of linear stability except for three mass ratios:

Accretion, ablation and propeller evolution in close millisecond pulsar binary systems

A model for the formation and evolution of binary millisecond radio pulsars in systems with low mass companions (<0.1 M_⊙) is investigated using a binary population synthesis technique. Taking into account the non conservative evolution of the system due to mass loss from an accretion disk as a result of propeller action and from the companion via ablation by the pulsar, the transition from the accretion powered to rotation powered phase is investigated. It is shown that the operation of the propeller and ablation mechanisms can be responsible for the formation and evolution of black widow millisecond pulsar systems from the low mass X-ray binary phase at an orbital period of ～0.1 day. For a range of population synthesis input parameters, the results reveal that a population of black widow millisecond pulsars characterized by orbital periods as long as ～0.4 days and companion masses as low as ～0.005 M_⊙ can be produced. The orbital periods and minimum companion mass of this radio millisecond pulsar population critically depend on the thermal bloating of the semi-degenerate hydrogen mass losing component, with longer orbital periods for a greater degree of bloating. Provided that the radius of the companion is increased by about a factor of 2 relative to a fully degenerate, zero temperature configuration, an approximate agreement between observed long orbital periods and theoretical modeling of hydrogen rich donors can be achieved. We find no discrepancy between the estimated birth rates for LMXBs and black widow systems, which on average are ${\sim}1.3\times10^{-5}~{\rm yr}^{-1}$ and $1.3\times10^{-7}~{\rm yr}^{-1}$ respectively.     

The perturbed ideal resonance problem

The author's previous studies concerning the Ideal Resonance Problem are enlarged upon in this article. The one-degree-of-freedom Hamiltonian system investigated here has the form $$\begin{array}{*{20}c} { - F = B(x) + 2\mu ^2 A(x)\sin ^2 y + \mu ^2 f(x,y),} \\ {\dot x = - F_y ,\dot y = F_x .} \\ \end{array}$$ The canonically conjugate variablesx andy are respectively the momentum and the coordinate, andμ ² is a small positive constant parameter. The perturbationf is o (A) and is represented by a Fourier series iny. The vanishing of ?B/?x≡B ⁽¹⁾ atx=x ₀ characterizes the resonant nature of the problem. With a suitable choice of variables, it is shown how a formal solution to this perturbed form of the Ideal Resonance Problem can be constructed, using the method of ‘parallel’ perturbations. Explicit formulae forx andy are obtained, as functions of time, which include the complete first-order contributions from the perturbing functionf. The solution is restricted to the region of deep resonance, but those motions in the neighbourhood of the separatrix are excluded.