The elliptical three-body problem. Triangular solution

The restricted three-body problem with eccentric orbit is reviewed and the positions of the triangular Lagrangian points (L₄, L₅) are determined. It is put in evidence the fact the fact L₄ and L₅ are situated at the corners of an isoscales triangle: AB = BC = 1 − e²/)1 + e cos ν )4/3 and AC = 1 − e²/)1 + e cos ν )     

Expansions of (r/a)m cos jv and (r/a)m sin jv to high eccentricities

Expansions of the functions (r/a)^cos jv and (r/a)m ^sin jv of the elliptic motion are extended to highly eccentric orbits, 0.6627 ... <e<1. The new expansions are developed in powers of (e–e*), wheree* is a fixed value of the eccentricity. The coefficients of these expansions are expressed in terms of the derivatives of Hansen's coefficients with respect to the eccentricity. The new expansions are convergent for values of the eccentricity such that |e–e*|<(e*), where the radius of convergence (e*) is the same of the extended solution of Kepler's equation. The new expansions are intrinsically related to Lagrange's series.     

Stationary equations of hydrogen excitation and ionization in prominences

The statistical equilibrium equations for the continuum and first 10 levels of a hydrogen atom show that the radiation of a bright prominence (the brightness of the H line has attained 56 mÅ of the disc centre spectrum) is completely due to scattering of the Sun radiation. The basic unknowns are separated with certainty: electron concentration (n _e = 3.0 × 10¹⁰ cm^–3), effective thickness (l = 4.2 × 10⁸ cm) and electron temperature (T _e = 5000 K).Radiation of a very bright prominence (A (H) = 213 mÅ; T _e = 7300 K; n _e = 5.0 × 10¹¹ cm^–3; l = 1.3 × 10⁷ cm) is on account of electron impacts (40%) and the Sun radiation scattering (60%).The parameters are shown to depend greatly on the prominence optical thickness in the lines of the first subordinate series of a hydrogen atom. In the course of determination all the parameters and 100 interconnected integral equations of the radiation diffusion have been thickness-averaged; the population of levels has been calculated by observations using the self-absorption factors.     

Apsidal Motion Study of the Eclipsing Binary Star V459 Cas

Photometric observation of the eccentric eclipsing binary V459 Cas (e = 0.0244) was performed to find a new rate of apsidal motion. Also the advance of the periastron is calculated theoretically by taking into account the Newtonian (classical) and general-relativistic effects according to physical and orbital parameters of the system. A new observed rate of apsidal motion of 19^∘.8/100 yr is computed which is not in agreement with the one reported earlier. Meanwhile a theoretical value of 2^∘.64/100 yr is obtained which is 7.49 times smaller than the observed one.     

Astrometric study of the relative motion of three stars with possible invisible companions based on homogeneous series obtained at Pulkovo with a 26-inch refractor

We investigate the relative motion of three stars, ADS 7446, 9346, and 9701, based on long-term observations with the Pulkovo 26-inch refractor. The relative motion of all three stars shows a perturbation that could be produced by the gravitational influence of an invisible companion. For ADS 7446, we have determined the orbit of the photocenter with a period of 7.9 yr; the mass of the companion is more than 0.4M _⊙. For ADS 9346, we have determined the radial velocities of the components: −14.60 km s⁻¹ for A and −13.94 km s⁻¹ for B. For ADS 9346 and 9701, we have determined the dynamical parallaxes, 24 and 20 mas, respectively, which are larger than those in the Hipparcos catalog by 5 mas, and calculated the orbits by the apparent motion parameter (AMP) method. The new orbit of ADS 9346 is: a = 5″.2, P = 2035 yr, and e = 0.46 at the system’s mass M = 2.5M _⊙. The new orbits of ADS 9701 are: (a = 2″.9, P = 829 yr, e = 0.54, M = 4.3M _⊙) and (a = 3″.8, P = 1157 yr, e = 0.53, M = 5.0M _⊙).     

The Eccentric Eclipsing Binary V889 Aquilae

Several new times of minimum light recorded with photoelectric or CCD means have been gathered for the eccentric eclipsing binary V889 Aql (P = 11.1 days, e = 0.37). Its O–C diagram is presented, and improved elements of the apsidal motion and the light-time effect (LITE) are given. We found a long apsidal motion period of about 24 400 ± 2400 years and a period of the third body of about 52 ± 2 years.     

Updated UBV Light-Curve and Period Analysis of Eclipsing Binary HS Herculis

UBV light-curves of the eclipsing binary HS Herculis, obtained in 2002–2003 observational seasons, were analysed with Wilson-Devinney computer code. New absolute dimensions of the system were calculated using the results of the light-curve analysis. Period variation of the system was also investigated. Several new times of minima have been secured for this problematic system. An apsidal motion with a period of 80.7 years was confirmed and a third body in a pretty eccentric orbit (e ₃ = 0.90 ± 0.08) with a period of 85.4 years was found. The corresponding internal structure constants of the binary system, log k ₂, and the mass of the third body were derived.     

Theoretical Ne v emission line ratios compared to solar observations

The recent level population calculations for Ne v by Aggarwal are used to determine the theoretical emission line ratios R ₁ = I(2s2p ^{3 1}D^o - 2s²2p^{2 1}D^e)/I(2s2p^{3 3}D ₂ ⁰ - 2s²2p^{2 3}P ₁ ^e ) and R ₂ = I(2s2p ^{3 1}D^o-2s²2p^{2 1}D^e)/I(2s2p ^{3 3}D ₃ ⁰ -2s²2p^{2 3}P ₂ ^e ). A comparison of these with observational data for a solar flare and erupting prominence obtained with the NRL XUV spectrograph on board Skylab reveals that R ₁ and R ₂ are in their predicted high density limits. Although the ratios cannot be used as density diagnostics for values of n _e typical of the solar transition region, it is shown that they are temperature sensitive and hence may be employed to determine the electron temperatures of Ne v line emitting regions.     

Interstellar electron density

We impose the requirement that the spatial distribution of pulsars deduced from their dispersion measures using a model of the galactic electron density (n _e ) should be consistent with cylindrical symmetry around the galactic centre (assumed to be 10 kpc from the Sun). Using a carefully selected subsample of the pulsars detected by the II Molonglo Survey (II MS), we test a number of simple models and conclude that (i) the effective mean 〈n_e〉) for the whole galaxy is 0.037_-0.012 ^+0.020 cm^-3, (ii) the scale height of electrons is greater than 300 pc and probably about 1 kpc or more, and (iii) there is little evidence for variation of n_e with galactic radius R_GC for R_Gc ≳ 5 kpc. Further, we make a detailed analysis of the contribution to n_e from H II regions. Combining the results of a number of relatively independent calculations, we propose a model for the galactic electron density of the formn _e (z) = 0.030 + 0.020 exp (- |z|/70) cm^-3 where z(pc) is the height above the galactic plane and the second term describes the contribution from H II regions. We believe the statistical uncertainties in the parameters of this model are quite small.     

On the perturbation of the anomalistic period of a highly eccentric orbit satellite due to the zonal harmonics

In this note a simple formula is given for the perturbation of the anomalistic period of a highly eccentric orbit due to the zonal harmonics. This perturbation depends essentially only on the semi-major axisa, the eccentricitye (or pericentre radius r =a(1-e)) and the latitude of the pericentre.     

Solving Kepler’s Equation using Bézier curves

This paper presents a non-iterative approach to solve Kepler’s Equation, M = E − e sin E, based on non-rational cubic and rational quadratic Bézier curves. Optimal control point coordinates are first shown to be linear with respect to orbit eccentricity for any eccentric anomaly range. This property yields the development of a piecewise (e.g., 3, 4) solving technique providing accuracies better than 10⁻¹³ degree for orbit eccentricity e ≤ 0.99. The proposed method does not require large pre-computed discretization data, but instead solves a cubic/quadratic algebraic equation and uses a single final Halley iteration in only a few lines of code. The method still provides accuracies better than 10⁻⁵ degree for the near parabolic worst case (e = 0.9999) with very small mean anomalies (M < 0.0517 deg). The complexity of the proposed algorithm is constant, independent of the parameters e and M. This makes the method suitable for extensive orbit propagations. Presented at the 7th Dynamics and Control of Systems and Structures in Space Conference, July 18–22, 2006, Greenwich, England.     

Light-time orbit and apsidal motion of the eclipsing binary U Ophiuchi

A comprehensive period study of the times of minima observed from 1881 to 1985 has been performed. Previous interpretations of the O–C diagram based on light-time effect are confirmed. The light-time orbit of U Oph has been revised using a differential corrections procedure. We get an eccentric orbit withe=0.22±0.06,P=38.7±0.2 yr, and a mass function . In addition, our analysis has revealed short-period apsidal motion (U/P=4515±75) in a slightly eccentric close orbit (e=0.0031±0.0003), allowing a reliable determination of the density concentration coefficient,k ₂=0.0059±4. A comparison with stellar evolutionary models calculated by Jeffery (1984) yields the helium contentY=0.28±0.05 and an age of 3×10⁷ yr for the components of U Oph.     

Analysis of the chromospheric hydrogen spectrum at the 1962 eclipse

Slitless spectrograms of the chromosphere obtained during the eclipse of 4–5 February 1962 have been analyzed to obtain the decrements of the level populations of hydrogen, the self-absorption in the Balmer lines, and parameters useful in construction of models of the low chromosphere.The decrement of the high energy levels of hydrogen inferred under the optically thin assumption does not vary significantly with height, and it appears to be unnecessary to seek large deviations from local thermodynamic equilibrium in the high levels. The observed Balmer-to-Paschen line intensity ratios have been used to infer self-absorption and opacities in the Balmer lines. The resulting population of the second energy level is about an order of magnitude smaller than that found by Athay and Thomas from the 1952 data.The chromospheric continuum was generally underexposed; the absence of observed continuum in the visible region of the spectrum made it impossible to derive a unique model from the 1962 data alone. However, the high Balmer line data and new theoretical solutions of the statistical equilibrium equations for hydrogen combined with corrected 1952 observations at 4700 A are compatible with a model having approximately the same temperature and neutral hydrogen structure as the 1952 model by Pottasch and Thomas but half the electron density: T _e = 6200K, N ₁ = 7.4 × 10¹³ cm^-3, N _e = 2.3 × 10¹¹ cm^-3 at 500 km and T _e = 7200K, N ₁ = 2.6 × 10¹² cm^-3, N _e = 1.7 × 10¹¹ cm^-3 at 1000 km.Based in part on a Ph.D. thesis submitted to the Department of Astro-Geophysics, University of Colorado.Now at the Department of Astronomy, Indiana University.     

A model cosmology based on gravity-electromagnetism unification

In this article the GEM (Brandenburg, 1992; Brandenburg, 1988) theory is applied to the problem of the cosmos in which most of the matter is hydrogen, spacetime is flat, and a Cosmic Background Radiation CBR field exists. Using the two postulates of the GEM theory: 1. That gravity fields are equivalent to an array ofE ×B drifts or a spacially varying Poynting field, such that spacetime is determined by EM fields so that the stress tensor of ultrastrong fields is self-canceling; 2. That EM and gravity fields and protons and electrons are unified at the Planck scale of lengths and energies and split apart to form distinct fields and separate particles at the Mesoscale of normal particle rest energies and classical radii. A new derivation is made of the formula forG found previously:G =e ²/(m _p m _e ) exp(-2R ^1/2) = 6.668 × 10^–8 dynes cm² g^–2wherem _p andm _e are the proton and electron masses respectively,R =m _p /m _e and is the fine structure constant, shows that quantum processes may occur which make the vacuum unstable to appearance of hydrogen thus allowing matter creation and a steady state universe to occur. The value for the Hubble Time calculated from this model isT _o = (3/((2)(R ^1/2)⁴))^1/3(r _e /c)(e ²/Gm _p m _e )= 19 Gyr wherer _e =e ² / (m _e c ²)and follows the form first hypothesized by Dirac(1937). The CBR is traced to this process of matter creation and its temperature is calculated as beingT _CBR = ((3/4)Gm _e ² c/( ² _o ))^1/4 = 2.66K where is the Thomson cross section of the electron and _o is the Stefan-Boltzman constant.     

Intensity and polarization line profiles in a semi-infinite Rayleigh-scattering planetary atmosphere. II. Variations of equivalent width over the disk

The study of the variation of equivalent width in a Rayleighscattering planetary atmosphere along the intensity equator and along the mirror meridian on whichμ =μ ₀ shows that the equivalent widths decrease monotonically towards the poles, the limb and the terminator with the following characteristics: (i) the weakest lines exhibit the maximum change; (ii) theI ^e _r component shows more change than theI ^e _r component; (iii) the decrease towards the limb or the terminator is not as sharp as that towards the poles; (iv)I ^e _r component shows more decrease towards the limb whileI ^e _r component shows more decrease towards the terminator; and (v) the relationW (μ, φ;μ ₀,φ ₀)= W (μ ₀,φ ₀;μ, φ) holds for the total intensity. These results are qualitatively in agreement with the observations of absorption bands in the spectra of Venus, Jupiter and Saturn.     

U - B - V photometry of the spectroscopic-visual triple HD 165590

HD 165590 is a visual binary (dG0 + dG5,P = 20 _. ^y 25,e = 0.96) whoseA component is an SB1 double (dG5 + dM:P = 0 _. ^d 88,e0.0). TheA pair (Aa +Ab) undergoes partial eclipses. PhotoelectricUBV photometry from Lines and one of the Automatic Photoelectric Telescopes, andV photometry from Scarfe are examined here. The data are from the 1977, 1984, 1985, and 1986 observing seasons. The non-eclipse light variations are analyzed with a FORTRAN program which does a sinusoidal curve fitvia least squares repeatedly to obtain the best period. Periods found from each observing season and passband are consistent with Boydet al. (1985): the greatest variations seem to be produced by a rotating (0 _. ^d 88), spotted, G0 star (theAa component). To the residuals from the first analysis a further curve fit is made to determine characteristics of the wave due to the ellipticity effect. An early limit on the spectral type of the unseenAb component, based on the primary eclpse depth and the upper limit on the depth of the unseen secondary eclipse, is K2. Eclipse depths and widths seen here suggest that theA pair's inclination = 74 _. ⁰ 9 ± 1⁰, close to theA +B inclination of 82 _. ⁰ 7 ± 2⁰ (Battenet al., 1979). TheA pair's orbital period does not appear to vary, appearing instead to be well-described by a new linear ephemeris (Hel. J.D. = 2443665.4568 + 0 _. ^d 8795045E) which does, however, take into account a variable light-travel-times as theA component orbits theA +B center of mass with a 20 _. ^y 25 period. The maximum light-travel-time O-C thus produced is + /–8 _. ^m 4 = + /–0 _. ^d 0059.     

Particle acceleration in impulsive solar flares

Second-step acceleration of nonrelativistic protons and ions in impulsive solar flares is discussed extending our earlier calculations for relativistic electrons. We derive the relevant particle transport equation, discussing in detail the influence of the particle's effective charge and mass number on the various momentum gain (stochastic acceleration, diffusive shock wave acceleration) and loss (Coulomb interactions, particle escape) processes. Analytical solutions for the ion-momentum spectra in the hard-sphere approximation are given. The inclusion of Coulomb losses modify the particle spectra significantly at kinetic energies smaller than E _B = 0.64( _e /5.0) MeV nucl.^–1 from the well-known Bessel function variation in long-duration flares. For equal injection conditions this modification explains the observed much smaller ion fluxes from impulsive flares at high energies as compared to long-duration flares. We also calculate the ³He/⁴He-isotope variation as a function of momentum in impulsive flares in the hard-sphere approximation and find significant variations near E _m = 0.38(T _e /2 × 10⁶ K) MeV nucl.^–1, where T _e is the electron temperature of the coronal medium.     

Large-Scale Structure of the Atmosphere of the Quiet Sun,Coronal Holes,and Plages as Deduced by Tomography Study

We present here the results of emission tomography studies, based on a new differential deconvolution method (DDM) of Laplace transform inversion, which we use for reconstruction of the coronal emission measure distributions in the quiet Sun, coronal holes and plage areas. Two methods are explored. The first method is based on the deconvolution of radioemission brightness spectra in a wide wavelength range (1 mm–100 cm) for temperature profile reconstructions from the corona to the deeper chromosphere. The second method uses radio brightness measurements in the cm–dm range to give a coronal column emission measure (EM).Our results are based on RATAN-600 observations in the range 2.0–32 cm supplemented by the data of other observatories during the period near minimum solar activity. This study gives results that agree with known estimates of the coronal EM values, but reveals the absence of any measurable quantities of EM in the transition temperature region 3 × 10⁴ –10⁵ K for all studied large-scale structures. The chromospheric temperature structure (T _e = 20,000–5800 K) is quite similar for all objects with extremely low-temperature gradients at deep layers.Some refraction effects were detected in the decimeter range for all Types of large-scale structures, which suggests the presence of dense and compact loops (up to N _e =(1–3)× 10⁹ cm^-3 number density) for the quiet-Sun coronal regions with temperature T _e > 5× 10^-5 K.     

Encounter in the keplerian field: Analytical treatment

In extending the results of Henon and Petit (1986) an algorithm is suggested for constructing the series representing the general encounter-type solution of the spatial eccentric Hill's problem. The series are arranged in powers of the eccentricity E of Hill's problem and two integration constants e and k characterizing eccentricity and inclination of the relative motion. A particular non-periodic solution of Henon and Petit corresponding to E = e = k = 0 is taken as an intermediary. The perturbations to this solution are constructed similar to the lunar theory of Hill and Brown.     

Recombination of electrons with H3+ and H5+ ions

The dissociative recombination coefficients α for capture of electrons by H₃⁺ and H₅⁺ ions have been determined as a function of electron temperature T_e using a microwave afterglow-mass spectrometer apparatus. At ion and neutral temperatures T_u+ = T_n = 240 K, the coefficient α (H₃⁺) is found to vary slowly with T_e at first, decreasing from 1.6 × 10^?7 cm³/s at T_e = 240 K to 1.2 × 10^?7 cm³/s at T_e = 500 K, thereafter falling as T_e^?1 over the range 500 K ? T_e, ? 3000 K. These results, which have a ± 20% uncertainty, agree satisfactorily over the common energy range (0.03–0.36 eV) with the recombination cross sections determined in merged beam measurements by Auerbach et al. At T₊ = T_n = 128 K, the coefficient α(H₅⁺) is found to be (1.8 ± 0.3) × 10^?6 [T_e(K)/300]^?0.69 cm³/s over the range 128 K ? T_e ? 3000 K, with a more rapid decrease, as T_e^?1, between 3000 K and 5500 K. The implications of these results for modelling planetary atmospheres and interstellar clouds are briefly touched on.