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.     

The ratios I(K1)/I(H1) and I(K3)/I(H3) of ca II as diagnostics of the chromosphere above sunspots

The ratios I(K ₁)/I(H ₁) and I(K ₃)/I(H ₃) were calculated from four semi-empirical models of sunspot umbra. We determined the dependencies of both ratios of such parameters as temperature gradient and atmospheric opacity. A certain influence on the expected ratios I(K ₁)/I(H ₁) and I(K ₃)/I(H ₃) can also come from the FIP effect provided it exists in the chromosphere above sunspot umbra. Theoretical and observed values of I(K ₁)/I(H ₁) and I(K ₃)/I(H ₃) are compared. It is shown that for one of the sunspots we observed, the values obtained for the ratio I(K ₁)/I(H ₁) cannot be explained in terms of existing umbra models.     

Evolution of relative helicity in active regions

The evolution dH _R/dt of relative helicity H _R provides a gauge-invariant measure of the helicity flow across the open surface S _o of an active region. With the incompressible approximation, reformulation of the evolution reveals that it is determined not only by the widely used cross-helicity h _mvp=A _pv contributed by the vector potential A _p of a reference potential field B _p, where v is the fluid velocity, but by another cross-helicity h _mvo=A _ov contributed by the vector potential A _o of the open field B _o in the region as well. Only under two conditions, (1) A _p=A (A is the transverse component of A _o), (2) v _z=0 (v _z is the longitudinal component of v) or A _z=0 (A _z is the longitudinal component of A _o), can h _mvo be merged into h _mvp to give the pioneering dH _R/dt equation shown in Equation (4) of Berger (1984). Results show that h _mvo originates from vh _o (h _o=A _oB _o is the helicity density of the open field) and should also be considered in dealing with the development of relative helicity in active regions. Finally, the equation to calculate dH _R/dt in active regions is synthesized and presented.     

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.     

The Schwabe and Gleissberg Periods in the Wolf Sunspot Numbers and the Group Sunspot Numbers

Three wavelet functions: the Morlet wavelet, the Paul wavelet, and the DOG wavelet have been respectively performed on both the monthly Wolf sunspot numbers (R_z) from January 1749 to May 2004 and the monthly group sunspot numbers (R_g) from June 1795 to December 1995 to study the evolution of the Gleissberg and Schwabe periods of solar activity. The main results obtained are (1) the two most obvious periods in both the R_z and R_g are the Schwabe and Gleissberg periods. The Schwabe period oscillated during the second half of the eighteenth century and was steady from the 1850s onward. No obvious drifting trend of the Schwabe period exists. (2) The Gleissberg period obviously drifts to longer periods the whole consideration time, and the drifting speed of the Gleissberg period is larger for R_z than for R_g. (3) Although the Schwabe-period values for R_z and R_g are about 10.7 years, the value for R_z seems slightly larger than that for R_g. The Schwabe period of R_z is highly significant after the 1820s, and the Schwabe period of R_g is highly significant over almost the whole consideration time except for about 20 years around the 1800s. The evolution of the Schwabe period for both R_z and R_g in time is similar to each other. (4) The Gleissberg period in R_z and R_g is highly significant during the whole consideration time, but this result is unreliable at the two ends of each of the time series of the data. The evolution of the Gleissberg period in R_z is similar to that in R_g.     

Pulsational instability of yellow hypergiants

Instability of population I (X = 0.7, Z = 0.02) massive stars against radial oscillations during the post-main-sequence gravitational contraction of the helium core is investigated. Initial stellar masses are in the range 65M _⊙ ≤ M _ZAMS ≤ 90M _⊙. In hydrodynamic computations of self-exciting stellar oscillations we assumed that energy transfer in the envelope of the pulsating star is due to radiative heat conduction and convection. The convective heat transfer was treated in the framework of the theory of time-dependent turbulent convection. During evolutionary expansion of outer layers after hydrogen exhaustion in the stellar core the star is shown to be unstable against radial oscillations while its effective temperature is T _eff > 6700 K for M _ZAMS = 65M _⊙ and T _eff > 7200 K for M _ZAMS = 90M _⊙. Pulsational instability is due to the κ-mechanism in helium ionization zones and at lower effective temperature oscillations decay because of significantly increasing convection. The upper limit of the period of radial pulsations on this stage of evolution does not exceed ≈200 day. Radial oscillations of the hypergiant resume during evolutionary contraction of outer layers when the effective temperature is T _eff > 7300 K for M _ZAMS = 65M _⊙ and T _eff > 7600 K for M _ZAMS = 90M _⊙. Initially radial oscillations are due to instability of the first overtone and transition to fundamental mode pulsations takes place at higher effective temperatures (T _eff > 7700 K for M _ZAMS = 65M _⊙ and T _eff > 8200 K for M _ZAMS = 90M _⊙). The upper limit of the period of radial oscillations of evolving blueward yellow hypergiants does not exceed ≈130 day. Thus, yellow hypergiants are stable against radial stellar pulsations during the major part of their evolutionary stage.     

Chemical Evolution of Gas-Rich Galaxy Mergers

We investigate numerically the chemodynamical evolution of major disc–disc galaxy mergers in order to explore the origin of the mass-dependent chemical, photometric and spectroscopic properties observed in elliptical galaxies. We investigate especially the dependence of the fundamental properties on merger progenitor disc mass (M _d). Three main results are obtained in this study:– More massive (luminous) ellipticals formed by galaxy mergers between more massive spirals have higher metallicity (Z) and thus show redder colours; the typical metallicity ranges from ∼ 1.0 solar abundance (Z∼ 0.02) for ellipticals formed by mergers with M _d = 10¹⁰ M _⊙to ∼ 2.0 solar (Z∼ 0.04) for those with M _d= 10¹² M _⊙.– Both the Mg₂ line index in the central part of ellipticals (R ≤ 0.1 R _e) and the radial gradient of Mg₂ (δ Mg₂ / δ log R) are more likely to be larger for massive ellipticals. δ Mg₂ / δ log R correlates reasonably well with the central Mg₂ in ellipticals. For most of the present merger models, ellipticals show a positive radial gradient of the H_β line index. – Both M/L _B and M/L _K (where M, L _B, and L _K are the total stellar mass of galaxy mergers, the B-band and the K-band luminosities, respectively) depend on galactic mass in such a way that more massive ellipticals have larger M/L _B and smaller M/L _K. This revised version was published online in July 2006 with corrections to the Cover Date.     

Interstellar Extinction and the Intrinsic Colors of Classical Cepheids in the Galaxy, the LMC, and the SMC

New methods are applied to samples of classical cepheids in the galaxy, the Large Magellanic Cloud, and the Small Magellanic Cloud to determine the interstellar extinction law for the classical cepheids, R _B:R _V:R _I:R _J:R _H:R _K= 4.190:3.190:1.884:0.851:0.501:0.303, the color excesses for classical cepheids in the galaxy, E(B-V)=-0.382-0.168logP+0.766(V-I), and the color excesses for classical cepheids in the LMC and SMC, E(B-V)=-0.374-0.166logP+0.766(V-I). The dependence of the intrinsic color (B-V)₀ on the metallicity of classical cepheids is discussed. The intrinsic color (V-I)₀ is found to be absolutely independent of the metallicity of classical cepheids. A high precision formula is obtained for calculating the intrinsic colors of classical cepheids in the galaxy: (<B>-<V>)₀=0.365(±0.011)+0.328(±0.012)logP.     

A New Method for Prediction of Peak Sunspot Number and Ascent Time of the Solar Cycle

The purpose of the present study is to develop an empirical model based on precursors in the preceding solar cycle that can be used to forecast the peak sunspot number and ascent time of the next solar cycle. Statistical parameters are derived for each solar cycle using “Monthly” and “Monthly smoothed” (SSN) data of international sunspot number (R _i). Primarily the variability in monthly sunspot number during different phases of the solar cycle is considered along with other statistical parameters that are computed using solar cycle characteristics, like ascent time, peak sunspot number and the length of the solar cycle. Using these statistical parameters, two mathematical formulae are developed to compute the quantities [Q _C] _n and [L] _n for each nth solar cycle. It is found that the peak sunspot number and ascent time of the n+1th solar cycle correlates well with the parameters [Q _C] _n and [L] _n /[S _Max] _n+1 and gives a correlation coefficient of 0.97 and 0.92, respectively. Empirical relations are obtained using least square fitting, which relates [S _Max] _n+1 with [Q _C] _n and [T _a] _n+1 with [L] _n /[S _Max] _n+1. These relations predict a peak of 74±10 in monthly smoothed sunspot number and an ascent time of 4.9±0.4 years for Solar Cycle 24, when November 2008 is considered as the start time for this cycle. Three different methods, which are commonly used to define solar cycle characteristics are used and mathematical relations developed for forecasting peak sunspot number and ascent time of the upcoming solar cycle, are examined separately.     

Study of nuclear decays during a solar eclipse: Thule Greenland 2008

A new prediction technique based on logarithmic values is proposed to predict the maximum amplitude (R _m) of a solar cycle from the preceding minimum aa geomagnetic index (aa _min). The correlation between lnR _m and lnaa _min (r=0.92) is slightly stronger than that between R _m and aa _min (r=0.90). From this method, cycle 24 is predicted to have a peak size of R _m(24)=81.7(1±13.2%). If the suggested error in aa (3 nT) before 1957 is corrected, the correlation coefficient between R _m and aa _min (r=0.94) will be slightly higher, and the peak of cycle 24 is predicted much lower, R _m(24)=52.5±13.1. Therefore, the prediction of R _m based on the relationship between R _m and aa _min depends greatly on the accurate measurement of aa.     

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.     

The S3–4 ionospheric irregularities satellite experiment: Probe detection of multi-ion component plasmas and associated effects on instability processes

The pulsed plasma probe technique has been expanded to include simultaneous determinations of absolute electron density, density fluctuations, electron temperature, and mean-ion-mass with resolution limited only by probe geometry, sheath size, and telemetry. The technique has been designed to test for coupling of electron density variations and ion composition irregularities in multi-component plasmas by the comparison of electron density fluctuation power spectraP _N(k) and a newly-developed diagnostic parameter, the mean-ion-mass fluctuation spectraM _i/M _iP _M(k). In addition, the experiment extends satellite-borne irregularity spectral analyses down to the 5–20 m range while attempting to identify F-region plasma instability processes on the basis of characteristics inN _e,T _e, N _e,P _N,M _i, andP _M. Initial results demonstrate the expanded diagnostic capability for high spatial resolution measurements of mean-ion-mass and provide experimental evidence for the role of ion composition in multi-stepped plasma instability processes. Specific results include a spectral indexX _n inP _N=A _nf^–X _n of 1.6–2.9 over the wavelength range from 1 km to 6 m under conditions identified with an unstable equatorial nighttime ionosphere. Simultaneous measurements ofM _i/M _i(P _M=A _M f ^–X _m) andN _e/N _e(P _N=A _n f ^–X _n) have shown a general behavior tending to lower power (A _m<A _n) and softer spectra (X _m<X _n) in ion mass fluctuations when compared with fluctuations in total plasma density. Limited analyses of the two power spectral elements raise hopes for the differentiation between plasma mechanisms that can lead to similar indices inP _N.Paper originally submitted to the journalSpace Science Instrumentation.     

The restricted planar isosceles three-body problem with non-negative energy

We consider a restricted three-body problem consisting of two positive equal masses m ₁ = m ₂ moving, under the mutual gravitational attraction, in a collision orbit and a third infinitesimal mass m ₃ moving in the plane P perpendicular to the line joining m ₁ and m ₂. The plane P is assumed to pass through the center of mass of m ₁ and m ₂. Since the motion of m ₁ and m ₂ is not affected by m ₃, from the symmetry of the configuration it is clear that m ₃ remains in the plane P and the three masses are at the vertices of an isosceles triangle for all time. The restricted planar isosceles three-body problem describes the motion of m ₃ when its angular momentum is different from zero and the motion of m ₁ and m ₂ is not periodic. Our main result is the characterization of the global flow of this problem.     

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).     

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.     

Perturbation equations of the elements of Vinti's intermediate orbit

Perturbation equations of the elements a, e, s, M_s, Ψ_s,Ω_sof Vinti's intermediate orbit are derived here correct to the second-order. Poisson terms have been eliminated from these equations.     

Methanol 72–81A+ emission and some maser series in Orion KL

Methanol 7₂–8₁ A ⁺ is mapped for the first time in Orion KL. Analysing the observed data and solving the statistical equilibrium and radiative transfer equations, it is concluded that line series ofJ ₂–(J+1)₁ A ⁺ (J=7,8,9) is in quasi-thermal emission rather than the masers in Orion KL. The maser spots of methanolJ ₂–J ₁ E (J=6,7) and 8₀–7₁ A ⁺ are distributed in the northeast part of the contour plot of 7₂–8₁ A ⁺. The physical conditions of the regions of maser seriesJ ₀–(J–1)₁ A ⁺ (J=7,8,9) are discussed. Also from the calculation results another maser seriesJ ₁–(J–1)₂ A ^– (J=10,11,12) that might coexist with maser seriesJ ₂–J ₁ E, is found. The sizes of the 2-dimension Gaussian fit plots of methanol 7₂–8₁ A ⁺ and HCOOCH₃ 10(0,10)–9(0,9)A are almost the same, and the main parts overlap each other.     

On the Aerodynamic Drag Force Acting on Interplanetary Coronal Mass Ejections

It is well known that the interaction of an interplanetary coronal mass ejection (ICME) with the solar wind leads to an equalisation of the ICME and solar wind velocities at 1 AU. This can be understood in terms of an aerodynamic drag force per unit mass of the form F _D/M=−(ρ_e AC _D/M)(V _i−V _e)∣V _i−V _e∣, where A and M are the ICME cross-section and sum of the mass and virtual mass, V _i and V _e the speed of the ICME and solar wind, ρ_e the solar wind density, C _D a dimensionless drag coefficient, and the inverse deceleration length γ=ρ_e A/M. The optimal radial parameterisation of γ and C _D beyond approximately 15 solar radii is calculated. Magnetohydrodynamic simulations show that for dense ICMEs, C _D varies slowly between the Sun and 1 AU, and is of order unity. When the ICME and solar wind densities are similar, C _D is larger (between 3 and 10), but remains approximately constant with radial distance. For tenuous ICMEs, the ICME and solar wind velocities equalise rapidly due to the very effective drag force. For ICMEs denser that the ambient solar wind, both approaches show that γ is approximately independent of radius, while for tenuous ICMEs, γ falls off linearly with distance. When the ICME density is similar to or less than that in the solar wind, inclusion of virtual mass effects is essential.     

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.