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).     

The regularizing function in resonance problems

The regularizing function ψ(x) in the theory of resonance removes the singularities discovered by Poincaré (1893), of the form 1/x, at theturning points x ₁ andx ₂ of thecritical argument x, librating in the rangex ₁≤x≤x ₂. This function has been explicitly introduced into the HamiltonianF ₀ of the Ideal Resonance Problem in the author's recent paper (1977) in order to remove the singularities in the second-order perturbations. It is shown here that this procedure can be extended toall orders. ThenF ₀ can be put into the form $$F_0 = B(y) + \Psi (x)$$ where ψ is thecomplete regularizing function, removing theclassical singularity of thesmall divisor, in addition to the singularities of Poincaré.     

Comparison of the classical and the global solutions of the Ideal Resonance Problem

The Ideal Resonance Problem is defined by the Hamiltonian $$F = B(y) + 2\varepsilon A(y) \sin ^2 x,\varepsilon \ll 1.$$ The classical solution of the Problem, expanded in powers of ε, carries the derivativeB′ as a divisor and is, therefore, singular at the zero ofB′, associated with resonance. With α denoting theresonance parameter, defined by $$\alpha \equiv - B'/|4AB''|^{1/2} \mu ,\mu = \varepsilon ^{1/2} ,$$ it is shown here that the classical solution is valid only for $$\alpha ^2 \geqslant 0(1/\mu ).$$ In contrast, the global solution (Garfinkelet al., 1971), expanded in powers ofμ=ε^1/2, removes the classical singularity atB′=0, and is valid for all α. It is also shown here that the classical solution is an asymptotic approximation, for largeα ², of the global solution expanded in powers ofα ^?2. This result leads to simplified expressions for resonancewidth and resonantamplification. The two solutions are compared with regard to their general behavior and their accuracy. It is noted that the global solution represents a perturbed simple pendulum, while the classical solution is the limiting case of a pendulum in a state offast circulation.     

Tame and chaotic behavior in the planar isosceles 3-body problem

We show that every planar isosceles solution of the three-body problem encounters a collision of the symmetric particles, either forwards or backwards in time. Regularizing analytically this collision, the solution has at least a syzygy configuration and/or leads to a total collapse. Some further simple results support the intuitive image on the tame local behavior of the motion as long as it does not lead to a triple collision. As a main result we prove that total collapse singularities, can be regularized in aC ¹-fashion with respect to time, for all values of the masses. Using symbolic dynamics, the chaotic character of theC ¹-regularized solutions is pointed out.     

Linear waves in a radiating and scattering grey medium

The equation of radiative acoustics is derived by taking into account the effect of a non-transverse magnetic field, and the solutions are schematically represented. The main results shown in Paper I and Paper II are valid even in the presence of a non-transverse magnetic field, and the only difference is that theadiabatic, isothermal, andisentropic speeds of sound and theradiation-acoustic speed in Paper I which respectively correspond to theadiabatic, isothermal, andisentropic magnetoacoustic speeds and theradiation-magnetoacoustic speed in Paper II are replaced by the sets of speeds ofadiabatic, isothermal, isentropic, andradiation-acoustic fast andslow waves, respectively.     

Structure and kinematics of the interstellar medium in the star-forming region in the BCD galaxy VII Zw 403 (UGC 6456)

The structure and kinematics of ionized supershells in the star-forming region in the BCD galaxy VII Zw 403 (UGC 6456) are analyzed using observations with the SCORPIO focal reducer on the 6-m Special Astrophysical Observatory telescope in three modes: direct imaging (in the Hα, [O III], and [S II] lines), long-slit spectroscopy, and spectroscopy with a scanning Fabry-Perot interferometer. In addition to the previously known bright H II regions and the faint giant ring that surrounds the entire starforming region, many new faint diffuse and arc structures have been detected. A fine structure of the giant ring has been revealed. We do not confirm the previously detected expansion of the bright shells around young stellar associations with a velocity of 50–70 km s^?1. We have estimated their expansion velocities to be no higher than 15–20 km s^?1; the corresponding kinematic age, no younger than 3–4 Myr, agrees well with the age of the compact OB associations associated with them. We correlate the faint extended filamentary and diffuse regions of ionized gas identified almost in the entire central region of the galaxy and the giant H II ring with the older (10 Myr) stellar population of the most recent starburst. Weak high-velocity [O III] and Hα line wings (up to 300 km s^?1 from the line center) have been detected in the brightest H II region. Such velocities have been observed in the galaxy for the first time. The previously published Hα luminosity measurements for the galaxy are refined.     

The cooling of a sunspot

This paper considers the recent criticism by Mullan (1973) of sunspot models and the cooling mechanism which I have proposed in Papers I, II and III of this series. The discussion of the cooling produced by an idealized flow cycle has been extended to include vertical temperature gradients which are consistent with a convectively unstable atmosphere. This leads to an expression for Mullan's parameter f (the ratio in which estimates of the energy flux based on an idealized Carnot cycle should be reduced) which is appropriate to this situation. It is shown that, for a cycle similar to that of Paper III, f = 0.82, while for one which has a vertical extent of order 5 Mm, f= 0.4. Hence the energy flux which, in principle, can be transported away from a sunspot by such a cycle is conservatively estimated to be 1.1 × 10²⁹ erg s^?1 compared with a typical sunspot energy deficit of 2.2 × 10²⁹ erg s^?1. Other criticisms relating to the magnetic field amplification and the ‘cool one’ model are discussed. It is concluded that the essential features of these models remain valid and that the modifications suggested by Mullan's criticism greatly increase their applicability to the sunspot problem.     

Eu III oscillator strengths and europium abundances in Ap stars

We present our calculations of the spectrum and oscillator strengths for the 4f⁷?(4f⁶5d+4f⁶6s) Eu III transitions. The calculations were performed with Cowan's RCN-RCG-RCE codes in the single-configuration approximation. A comparison of computed level lifetimes with experimental data for three levels shows that the scale of theoretical oscillator strengths could be overestimated by a factor of 3. The theoretical oscillator strengths of red Eu III lines are two orders of magnitude smaller than their astrophysical oscillator strengths derived by Ryabchikova et al. (1999) from the condition of ionization balance. The new oscillator strengths were tested by analyzing the Eu abundance using Eu II and Eu III lines in the spectra of hot peculiar stars (α² CVn is a typical representative) and cool peculiar stars (β CrB is a typical representative). First, we computed non-LTE corrections, which proved to be significant for α² CVn. We also analyzed the Eu II λ6645.11-Å line as well as ultraviolet and optical Eu III lines. We show that the new oscillator strengths together with the non-LTE corrections allow the contradiction between the Eu abundances derived by Ryabchikova et al. (1999) separately from optical Eu II and Eu III lines in α² CVn to be resolved. The new Eu abundance, log(Eu/N _tot)=?6.5, also faithfully describes the blended near-ultraviolet resonance Eu III lines. Using the new Eu III oscillator strengths to analyze the spectrum of the cool Ap star β CrB, we found a significant deviation of the n(Eu II)/n(Eu III) ratio from its equilibrium value. For a chemically homogeneous model atmosphere, to obtain the observed intensity of the Eu III λ 6666.35-Å line, the Eu abundance must be increased by two orders of magnitude compared to that required to describe the Eu II λ 6645.11-Å line. We discuss the possibility of explaining the observed intensities of Eu II and Eu III lines in the spectrum of β CrB by the presence of an inhomogeneous atmosphere with Eu concentrated in its uppermost layers. In such atmospheres, the role of non-LTE effects becomes dominant.     

Light and dark soils at the apollo 16 landing site

On the basis of inert gas systematics alone, the soilsnow near the surface at the Apollo 16 landing site can be divided into three major groups: Group I (North Ray Crater Soils), Group II (Light Soils), and Group III (Dark Soils). Only five soils do not fit this scheme. The inert gas-based classification is correlated with the chemistry of the soils. Group I soils are relatively poor in K, Fe, Ti and Zn, compared to Group II and III soils. The classification is also correlated with reflectivity. Group I and II soils are generally the light soils in the landing area, while the Group III soils are the dark soils. The groups are not randomly distributed in the landing area. Group I soils occur only at stations 11 and 13 on the ejecta blanket of North Ray Crater. Group II soils occur abundantly at stations 1 and 2, and in spots on Stone Mountain. Group III soils are abundant on Stone Mountain and at station 10. We suggest here that Group I soils are principally derived from the light friable unit, one of the three units inside North Ray Crater, as described by Ulrich. We suggest that Group II soils are mainly derived from the light matrix breccia unit. Group III soils are mixtures of materials from all three units. We conclude that soils with the properties of Group III soils have been at the surface continuously for long times. However, going backwards in time, these soils probably had increasingly larger (Ar⁴⁰/Ar³⁶)t ratios. The ejecta blanket of North Ray Crater is a temporary ‘anomaly’ in the landing site. However, soils with the properties of Group I soils, but with larger (Ar⁴⁰/Ar³⁶)t ratios may turn up in Apollo 16 core tubes. The Group II soils show a record of solar wind exposure in the distant past (i.e., they have relatively large Art ⁴⁰/ARt ³⁶ ratios). From this we conclude that the regolith at Apollo 16 contains sizeable ‘pockets’ or horizons at depth which are the sources of the Group II soils. The materials in these pockets may be akin to soil 61 220.     

The global solution of the problem of the critical inclination

The publication of the solution of the Ideal Resonance Problem (Garfinkelet al., 1971) has opened the way for a complete first-orderglobal theory of the motion of an artificial satellite, valid for all inclinations. Previous attempts at such a theory have been only partially successful. With the potential function restricted to $$V = - 1/r + J_2 P_2 (\sin \theta )/r^3 + J_4 P_4 (\sin \theta )/r^5 ,$$ the paper constructs aglobal solution of the first order in √J ₂ for the Delaunay variablesG, g, h, l and for the coordinatesr, θ, and ?. As a check, it is shown that this solution includes asymptotically theclassical limit with the critical divisor 5 cos² i?1. The solution is subject to thenormality condition $$eG^2 /(1 + \frac{{45}}{4}e^2 ) \geqslant O\left[ {\left| {\frac{1}{5}(J_2 + J_4 /J_2 )} \right|^{1/4} } \right],$$ which bounds the eccentricitye away from zero in deep resonance. A historical section orients this work with respect to the contributions of Hori (1960), Izsak (1962), and Jupp (1968).     

Bianchi type-III bulk viscous dust filled universe in Lyra geometry

Bianchi Type III bulk viscous dust filled cosmological models in Lyra geometry are investigated. To get the deterministic model of the universe, we have assumed two conditions: (i) ζθ=constant; and (ii) shear (σ) is proportional to the expansion (θ). This condition leads to B=C ⁿ where ζ the coefficient of bulk viscosity, θ the expansion in the model; B and C are metric potentials and n a constant. The physical and geometrical aspects of the model and singularities in the model are also discussed.     

Spectroscopic observations of the classical symbiotic star V1413 AQL at the phase of its transition to quiescence

We present the results of our spectroscopic and photometric observations of the classical symbiotic star V1413 Aql in 2003–2007, which cover the transition phase of its hot component from activity to quiescence. Various quiescence criteria for the hot component of the system are analyzed. Judging by its photometric characteristics, the system had not yet returned to quiescence by 2007, although a fairly strong He II λ4686 Å line observed previously during quiescence in 1993 appeared in its spectrum. We model the continuum energy distribution for V1413 Aql based on a standard three-component model and a model with an accretion disk. Analysis of the forbidden [O III] and [Ne III] lines shows that the neon abundance in V1413 Aql may be enhanced with respect to the oxygen abundance. The cool component of V1413 Aql has been found to be a variable M5-type red giant, with the variability amplitude at λ = 7500 Å being at least 2 ^m .     

Beta decay rates forS-process studies

The rates for a variety of beta decay processes have been determined as a function of temperature for nuclei which can participate in thes-process production of heavy elements, occurring in the presence of the²²Ne(α,n)²⁵ Mg neutron source operating in the convective helium shells of thermally pulsing stars. Specifically: calculated half-lives are presented for electron emission, positron emission, and electron capture over the temperature range 10⁸–10⁹ K.     

Period-luminosities relation of classical Cepheids

A new PL-relation (10) — Figure 2 for the Cepheids in the Galaxy, the Magellanic Clouds and M31 has been constructed. On deriving this relation both the period-radius (3) and period-colour relations (7), (8) and (9) are essentially used. The PC-relation (7) determined after the colours of 88 galactic cepheids (Table I), which are obtained from the colour-spectrum relation (6) — Figure 1, common for Cepheids and non-variable supergiants, are used also for the M31 Cepheids, whereas (8) and (9) are for the Large and Small MC Cepheids, respectively, all three PC-relations having a common slope. The comparison of the relations (8) and (9) with (7) shows that the LMC and SMC Cepheids are bluer than the galactic ones with 0^m.04 and 0^m.19, respectively, probably because of their metal-poor abundance. The places of thes-Cepheids in Figure 2 show that these Cepheids possess a dissimilar PL-relation with a different slope. The reason for such a difference is that thes-Cepheids are first harmonic pulsators. The distance moduli of the three galaxies under discussion are obtained from the PL-relation (10). The colour-coefficient of period-colour-luminosity-relation is briefly discussed. The general conclusion based on a comparison of the PL-relation in the present paper with those by other authors (Table V) is that our PL-relation differs in the zero-point by less than 0^m.2; therefore, the manner of constructing the PL-relation by means of PR and PC-relations is reasonable and useful.     

The structure and microphysical properties of the Venus clouds: Venera 9, 10, and 11 data

Data processing and interpretation of the nephelometer measurements made in the Venus atmosphere aboard the Venera 9, 10 and 11 landers in the sunlit hemisphere near the equator are discussed. These results were used to obtain the aerosol distribution and its microphysical properties from 62 km to the surface. The main aerosol content is found in the altitude range between 62 km (where measurements began) and 48 km, the location of the cloud region. Three prominent layers labeled as I (between 62 and 57 km), II (between 57 and 51 km) and III (between 51 and 48 km), each with different particle characteristics are discovered within the clouds. The measured light-scattering patterns can be intrepreted as having been produced by particles with effective radii from 1 to 2 μm depending on height and indices of refractivity from 1.45 in layer I to 1.42 in layer III. These values do not contradict the idea that the droplets are made of sulfuric acid. In layers II and III the particle size distribution is at least bimodal rather than uni-modal. The index of refraction is found to decrease to 1.33 in the lower part of layer II, suggesting a predominant abundance of larger particles of different chemical origin, and chlorine compounds are assumed to be relevant to this effect. In the entire heightrange of the Venera 9–11 craft descents, the clouds are rather rarefied and are characterized by a mean volume scattering coefficient σ ～ 2 × 10^?5 cm^?1 that corresponds to the mean meteorological range of visibility of about 2 km. The average mass content of condensate is estimated to be equal to 4 × 10^?9 g/cm³, and the total optical depth of clouds to τ ～ 35. Near the bottom of layer III clouds are strongly variable. In the subcloud atmosphere a haze was observed between 48 and 32 km; that haze is mainly made of submicron particles, r_eff ～ 0.1μm. The atmosphere below that is totally transparent but separate (sometimes possibly disappearing) layers may be present up to a height of 8 km above the surface. A model of this region with a very low particle density (N ? 2–3 cm^?3) strongly refractive large particles (r_eff ? 2.5 μm; 1.7 < n < 2.0) provided satisfactory agreement. The optical depth of aerosol in the atmosphere below the subcloud haze does not exceed 2.5.     

New in the optical spectrum and kinematic state of the atmosphere of the variable V1027 Cyg (=IRAS 20004+2955)

Based on our high-spectral-resolution observations performed with the NES echelle spectrograph of the 6-m telescope, we have studied the peculiarities of the spectrum and the velocity field in the atmosphere and envelope of the cool supergiant V1027 Cyg, the optical counterpart of the infrared source IRAS 20004+2955. A splitting of the cores of strong absorptions of metals and their ions (Si II, Ni I, Ti I, Ti II, Sc II, Cr I, Fe I, Fe II, BaII) has been detected in the stellar spectrum for the first time. The broad profile of these lines contains a stable weak emission in the core whose position may be considered as the systematic velocity V _sys = 5.5 km s^?1. Small radial velocity variations with an amplitude of 5–6 km s^?1 due to pulsations have been revealed by symmetric low- and moderate-intensity absorptions. A long-wavelength shift of the Hα profile due to line core distortion is observed in the stellar spectrum. Numerous weak CN molecular lines and the KI 7696 Å line with a P Cyg profile have been identified in the red spectral region. The coincidence of the radial velocities measured from symmetric metal absorptions and CN lines suggests that the CN spectrum is formed in the stellar atmosphere. We have identified numerous diffuse interstellar bands (DIBs) whose positions in the spectrum, V _r (DIBs) = ?12.0 km s^?1, correspond to the velocity of the interstellar medium in the Local Arm of the Galaxy.     

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.     

Turbulence,convection, and mixing in red giant stars: Some empirical approaches based on high resolution spectroscopy

The differential velocity field in cool stars can be measured effectively on photographic plates by the use of a PDS micro-densitometer, from which it is shown that radial velocity gradients are larger for stars with larger turbulent velocities, which are determined from the high resolution echellograms. This indicates that the stellar turbulence may have something to do with the differential velocity field in stellar atmospheres. As an observational probe of the mixing processes in red giant stars, stellar abundances that are sensitive to mixing are determined on the basis of high resolution Fourier Transform spectroscopy. For example, the¹²C/¹³C ratio has now been determined for large number of red giant stars and shows characteristic changes through the first red giant branch to the asymptotic giant branch, including both the pre-thermal pulsing and the thermal pulsing phases. This information, together with additional information on the¹⁶O/¹⁷O ratio and on CNO abundances, provides useful constraints on the theory of mixing in red giant stars.Paper presented at the IAU Third Asian-Pacific Regional Meeting, held in Kyoto, Japan, between 30 September-6 October, 1984.     

The structure of dust tails of comets II. The tail and dust content of comet Arend-Roland

The modified distribution function of dust particles, f(γ), which can be determined from tail brightness profiles on the basis of mechanical theory, is discussed with special regard to its reliability and accuracy. Physical significance of f(γ) is also discussed in terms of dust model parameters, and it is shown that f(γ), if treated carefully, will serve as an effective tool in studying cometary dust. Four isophotes of Comet Arend-Roland, 1957 III, in the orange-red light (λλ 0.53–0.68 micron) obtained by Ceplecha (1958), are analysed by the numerical method described in Paper I (KIMURA and LIU 1975) with some improvements and higher approximations. The distribution f(γ) thus obtained shows a bimodal character with peaks at γ = 0.10 and 0.010 with a relative height ratio of 1 to 0.6. Dust emission rate, which is assumed to follow the inverse square law of heliocentric distance, is estimated to give P_dC_sca = (1.3±0.5) × 10⁹ cm²/sec, where P_d is the rate of particle emission at 1 a.u. and the C_sca is a mean effective cross section of particles for light scattering including the phase effect (the scattering angles of present interest range from about 80 to 100 degrees).