A spectrographic study of R Canis Majoris

New spectroscopic elements of R Canis Majoris are given in this paper, computed from 17 plates secured at the Astronomical Observatory of Merate. A new orbit has also been computed with the program by Bertiau, from 81 observations covering a period of about 40 years.Two models of this binary system are proposed, based on two different assumptions and on a newly determined mass function. It is confirmed that the main peculiarity of this binary with regard to the mass-luminosity relation is real; there is also an evidence of mass loss from the Lagrangian pointL ₂ as suggested by Kitamura.A suspected variation of the velocityV ₀ of the center of mass seems to indicate the presence of a third body. The evidnece is, however, not yet conclusive.     

General time elements for the two body problem

When integrating a perturbed two-body problem, very often the propagation of the numerical error is reduced by using a new time variables defined by dt/ds=|q| ⁿ , (|q| is the radial distance,t the time). This paper introduces a time element for such transformations, i.e., a new variablet _n is defined so that dt _n/ds=1+ (perturbing terms) andt=F _n(t_n), whereF _n is a known function. The time element equation should be useful in reducing the error in the determination of the timet.F _n is given explicitly forn=1, 3/2, 2, 5/2 and 3, and a general expression is given for other values.The work was performed while the author was an NRC Senior Research Associate, Goddard Space Flight Center, Greenbelt, Md., U.S.A.     

Report on the dynamical evolution of an axially symmetric quasar model

The role of the angular momentum in the regular or chaotic character of motion in an axially symmetric quasar model is examined. It is found that, for a given value of the critical angular momentumL _zc , there are two values of the mass of the nucleusM _n for which transition from regular to chaotic motion occurs. The [L _zc – M _n ] relationship shows a linear dependence for the time independent model and an exponential dependence for the evolving model. Both cases are explained using theoretical arguments together with some numerical evidence. The evolution of the orbits is studied, as mass is transported from the disk to the nucleus. The results are compared with the outcomes derived for galactic models with massive nuclei.     

On the adjustment of outer solar layer models

Using the electron density n _e as an independent variable agreement between the models of the convective zone, photosphere, chromosphere, corona and solar wind is obtained. As a base the known data about the mean models of the individual layers of the quiet Sun are taken (i.e. without taking account of inhomogeneities and deviations from spherical symmetry). The chromospheric region is the exclusion. Here the run of T _e (n _e ) is revised anew to provide a smooth transition from temperature minimum to the corona and to satisfy the observed intensity distribution in the shortwave radio emission spectrum.A plot of the gas density versus n _e permits to get a clear representation about the rate of change of the degree of ionization x and to evaluate quickly the numerical values of x.     

On the A-transform of the exponential integrals

In this paper, a technique of recursive analysis is developed for the integral transform A of the exponential integral functionsE _n which is denoted as _n (). The main result of this analysis enables us to establish a two-term recurrence formula for _n (0) and a three-term recurrence formula for _n (); 0. A computational algorithm based on these formulae is also constructed and its numerical results forn=2(1)25 are presented to 15-digit accuracy.     

General integral transform of the exponential integrals

General integral transform of the exponential integralsE _n is considered and will be denoted asB _(k) ⁿ (). Different expressions and the equations satisfied byB _(k) ⁿ are developed. Two-term recurrence formula forB _(k) ⁿ (0) and three-term recurrence formula forB _(k) ⁿ (); 0 will be established for a givenk1 andn=2,3, ...,N. The computational algorithms based on these formulae are also constructed for the casesk=1,2,3, andn2. Finally the numerical results fork=2,3 andn=2(1)25 are presented to 15-digit accuracy     

A direct numerical approach to the Chandrasekhar'sH-functions for arbitrary characteristic functions

The purpose of this paper is to present a numerical technique to directly compute the Chandrasekhar'sH ()-function for anisotropic scattering in terms of the roots of the characteristic equations as well as the quadrature points of a certain degreen employed to approximate the definite integral involved in the basic equation. The principal feature of the algorithm proposed here is a compact computer code to enumerate _n C _m combinations ofn distinct integers {1,...,n} takenm at a time. With these quantities available, the coefficients of the polynomial equation of the characteristics equation can be readily computed for any given characteristic function, so that a standard technique such as the Laguerre method can be applied to find all the roots.It is shown that the results obtained for some representativeH()-functions using the present technique with relatively low-order formula (e.g.,n=7) are sufficiently accurate for all practical purposes.     

Expansion theory for the elliptic motion of arbitrary eccentricity and semi-major axis

In this paper of the series, literal analytical expressions for the coefficients of the Fourier series representation ofF will be established for anyx _i ; withn, N positive integers 1 and | _i | fori=1, 2,...n. Moreover, the recurrence formulae satisfied by these coefficients will also be established. Illustrative analytical examples and a full recursive computational algorithm, with its numerical results, are included. The applications of the recurrence formulae are also illustrated by their stencils. As a by-product of the analyses is an integral which we may call a complete elliptic integral of thenth kind, in which the known complete elliptic integrals (1st, 2nd and 3rd kinds) are special cases of it.     

Understanding the chemical complexity in Circumstellar Envelopes of C-Rich AGB stars: the case of IRC +10216

The circumstellar envelopes of carbon-rich AGB stars show a chemical complexity that is exemplified by the prototypical object IRC +10216, in which about 60 different molecules have been detected to date. Most of these species are carbon chains of the type C _n H, C _n H₂, C _n N, HC _n N. We present the detection of new species (CH₂CHCN, CH₂CN, H₂CS, CH₃CCH and C₃O) achieved thanks to the systematic observation of the full 3 mm window with the IRAM 30m telescope plus some ARO 12m observations. All these species, known to exist in the interstellar medium, are detected for the first time in a circumstellar envelope around an AGB star. These five molecules are most likely formed in the outer expanding envelope rather than in the stellar photosphere. A pure gas phase chemical model of the circumstellar envelope is reasonably successful in explaining the derived abundances, and additionally allows to elucidate the chemical formation routes and to predict the spatial distribution of the detected species.     

Expansion theory for the elliptic motion of arbitrary eccentricity and semi-major axis

In this paper of the series, literal analytical expressions for the coefficients of the Fourier series representation ofG will be established for anyx _i; withn, N positive integers and |_i|<1 fori=1, 2, ... n. Moreover, the recurrence formulae satisfied by these coefficients will also be established. Illustrative analytical examples and a full recursive computational algorithm, with its numerical results, are included. The applications of the recurrence formulae are also illustrated by their stencils. As by-products of the analyses are two important periodic integrals developed analytically and computationally.     

Complete Zonal Problem of the Artificial Satellite: Generic Compact Analytic First Order in Closed Form

This paper is a contribution to the Theory of the Artificial Satellite, within the frame of the Lie Transform as canonical perturbation technique (elimination of the short period terms). We consider the perturbation by any zonal harmonic J _n (n ≥ 2) of the primary on the satellite, what we call here the complete zonal problem of the artificial satellite. This is quite useful for primaries with symmetry of revolution. We give an analytical formula to compute directly the first order averaged Hamiltonian. The computation is carried out in closed form for all terms, avoiding therefore tedious expansions in the eccentricity or in any anomaly; this feature makes the averaging process, not only valid for all kind of elliptic trajectories but at the same time it yields the averaged Hamiltonian in a very short and compact way. The formula allows us to now skip the averaging process, which means an asymptotic gain of a factor 3n/2 regarding the computational cost of the n ^th zonal. Our analytical formulae have been widely checked, by comparison on one hand with published works (Brouwer, 1959) (which contained results for particular zonal harmonics, let’s say typically from J ₂ to J ₈), and on the other hand with the results of 3 symbolic manipulation software, among which the MM (standing for ‘Moon’s series Manipulator’), which has already been used and described in (De Saedeleer B., 2004). Additionally, the first order generator associated with this transformation is given into the same closed form, and has also been validated.     

Cosmic Ray Production Rates in Supernova Remnants

Supernova Remnants (SNRs) are the most likely sources of the galactic cosmic rays up to energies of about 10¹⁵ eV/nuc. The large scale shock waves of SNRs are almost ideal sites to accelerate particles up to these highly non-thermal energies by a first order Fermi mechanism which operates through scattering of the particles at magnetic irregularities. In order to get an estimate on the total amount of the explosion energy E _SNconverted into high energy particles the evolution of a SNR has to be followed up to the final merging with the interstellar medium. This can only be done by numerical simulations since the non-linear modifications of the shock wave due to particle acceleration as well as radiative cooling processes at later SNR stages have to be considered in such investigations. Based on a large sample of numerical evolution calculations performed for different ambient densities n _ext, SN explosion energies, magnetic fields etc. we discuss the final ‘yields’ of cosmic rays at the final SNR stage where the Mach number of the shock waves drops below 2. At these times the cosmic rays start to diffuse out of the remnant. In the range of external densities of10^-2 ≤ n _ext/[cm^-3] ≤ 30 we find a the total acceleration efficiency of about 0.15 E _SN with an increase up to 0.24 E _SN at maximum for an external density of n _ext = 10 cm^-3. Since for the larger ambient densities radiative cooling can reduce significantly the total thermal energy content of the remnant dissipation of Alfvén waves can provide an important heating mechanism for the gas at these later stages. From the collisions of the cosmic rays with the thermal plasma neutral pions are generated which decay subsequently into observable γ-rays above 100 MeV. Hence, we calculate these γ-ray luminosities of SNRs and compare them with current upper limits of ground based γ-raytelescopes. The development of dense shells due to cooling of the thermal plasma increases the γ-ray luminosities and e.g. an external density of n _ext = 10 cm^-3 with E _SN = 10⁵¹ erg can lead to a γ-ray flux above 10^-6 ph cm^-2 s^-1 for a remnant located at a distance of 1 kpc. This revised version was published online in July 2006 with corrections to the Cover Date.     

High‐mass resolution molecular imaging of organic compounds on the surface of Murchison meteorite

High‐resolution mass spectrometry (HRMS) imaging by desorption electrospray ionization (DESI) coupled with Orbitrap MS using methanol (MeOH) spray was performed on a fragment of the Murchison (CM2) meteorite in this study. Homologues of C_nH_2n–1N₂⁺ (n = 7–9) and C_nH_2nNO⁺ (n = 9–14) were detected on the sample surface by the imaging. A high‐performance liquid chromatography (HPLC)/HRMS analysis of MeOH extracts from the sample surface after DESI/HRMS imaging indicated that the C_nH_2n–1N₂⁺ homologues corresponds to alkylimidazole, and that a few isomers of the C_nH_2nNO⁺ homologues present in the sample. The alkylimidazoles and C_nH_2nNO⁺ homologues displayed different spatial distributions on the surface of the Murchison fragment, indicating chromatographic separation effects during aqueous alteration. Moreover, the distribution pattern of compounds is also different among homologues. This is probably also resulting from the separation of isomers by similar chromatographic effects, or different synthetic pathways. Alkylimidazoles and the C_nH_2nNO⁺ homologues are mainly distributed in the matrix region of the Murchison by mineralogical observations, which is consistent with previous reports. Altered minerals (e.g., Fe‐oxide, Fe‐sulfide, and carbonates) occurred in this region. However, no clear relationship was found between these minerals and the organic compounds detected by DESI/HRMS imaging. Although this result might be due to scale differences between the spatial resolution of DESI/HRMS imaging and the grain size in the matrix of the Murchison, our results would indicate that alkylimidazoles and the C_nH_2nNO⁺ homologues in the Murchison fragment were mainly synthesized by different processes from hydrothermal alteration on the parent body.     

General precession computed by numerical integration of ordinary differential equations

It is suggested that the effects of change in the celestial coordinates due to general precession be computed by solving numerically the differential equations for the instataneous rates of change in right ascension and declination. An approximate solution in closed from is given. Similar differential equations may be solved for the complete rotation matrix, requiring only the knowledge of the variations of the two function m and n with time. This suggests the existence of a relation between the classical precessional elements z, ζ₀ and Θ, and this relation is derived. Some numerical examples are also presented.     

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.     

Dynamical theory of viscous tides in close binary systems

The aim of the present paper has been to investigate quantitative aspects of the phenomenon of tidal lag in close binary systems, the components of which rotate (in a direct or retrograde sense) in periods which differ from that of orbital revolution. The components constituting the binary are regarded as self-gravitating configurations, consisting of viscous compressible fluid, the viscosity of which varies with the 2.5th power of local temperature (indicated by theoretical investigations of the viscosity of hydrogen plasma). The equilibrium structure of the components has been assumed to be polytropic of indexn; and numerical computations were carried out for the values ofn=1.5, 2.5, 3.5, and 4.5. The magnitudes of the tidal lag /2 – _i for these models and for different values of the ratio of the angular velocities of rotation and revolution are listed in Tables III–XLII in terms of six values of a non-dimensional parameterZ which is proportional to viscosity.     

Populations of Rydberg levels of atoms in the interstellar medium

Populations of Rydberg levels of atoms are calculated including effects of lowering the atomic ionization potential. Due to these effects the boundary conditions in b_n problem is lim b_n = 0. The b_n factors were computed for T = 100 K. N_e = 0.1 cm⁻³ with b_n = 0 when n ≧ 3000. The result allows to interpret observational data of the carbon decametric lines without the use of the subthermal dielectronic recombination mechanism. The calculations predict high relative intensities of C_nβ and C_nγ lines.     

On the number of central configurations in the N-body problem

Central configurations are critical points of the potential function of the n-body problem restricted to the topological sphere where the moment of inertia is equal to constant. For a given set of positive masses m ₁,..., m _n we denote by N(m ₁, ..., m _n, k) the number of central configurations' of the n-body problem in ^k modulus dilatations and rotations. If m _{n 1},..., m _n, k) is finite, then we give a bound of N(m ₁,..., m _n, k) which only depends of n and k.     

Neutralization and stabilization of particle streams in the corona and type III radio bursts

The processes by which streams of charged particles become charge and current neutralized in the corona are investigated. It is shown that a large amplitude plasma wave, which is related to precursor phenomenon in type III bursts and possibly plasma radiation from type IV bursts, will be excited at the head of the stream. The energy extracted from the stream to produce this plasma wave is computed and used to set conservative upper limits on the densities of possible excitors for type III bursts. For electron streams the density n _s < 10^–5 n _e, where n _e is the density of the background plasma. For proton streams n _s < 1.8 × 10^–2 n _e. The energy extracted from the stream is also used to set upper limits on the lifetimes of relativistic electrons stored in the corona and it is concluded that for n _e > 10² cm^–3 this loss must be taken into account. Since electron streams cannot produce their own stabilizing ionacoustic waves because they would violate the condition n _s < 10^–5 n _e, other mechanisms for producing ion-acoustic waves in the corona are examined. Another stabilization mechanism due to velocity inhomogeneity is investigated.The National Center for Atmospheric Research is sponsored by the National Science Foundation.