Some expansions of the elliptic motion to high eccentricities

Some classic expansions of the elliptic motion — cosmE and sinmE — in powers of the eccentricity 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 are given in terms of the derivatives of Bessel functions with respect to the eccentricity. The expansions have the same radius of convergence (e*) of the extended solution of Kepler's equation, previously derived by the author. Some other simple expansions — (a/r), (r/a), (r/a) sinv, ..., — derived straightforward from the expansions ofE, cosE and sinE are also presented.     

Population I pulsating stars

Evolutionary masses corresponding to various evolutionary phases of Population I pulsating stars (89 Delta Scuti variables and 155 classical cepheids) are interpolated in the systems of tracks of Iben (1967) and Paczyski (1970). The evolutionary masses are larger in the latter system than in the former one. The uncertainty of the evolutionary mass of a star is estimated, when various evolutionary phases are possible for this star (a smaller evolutionary mass corresponds to a later phase). Semi-empirical period-evolutionary mass-colour (P-M _e -C) and period-evolutionary mass (P-M _e ) relations are derived for various modes, groups of stars, colour indices (and effective temperature), and evolutionary phases. For Delta Scuti stars, the uncertainty of evolutionary masses calculated from theP-M _e relations for different modes, is estimated. The improvement of the evolutionary mass accuracy is estimated, when aP-M _e -C relation is used instead of the correspondingP-M _e relation. The theoretical and semi-empirical period ratios of radial pulsations derived from theP-M _e relations for Delta Scuti stars, are compared. There is a relatively good agreement between theP-M _e relations for the two types of Population I pulsating stars, but a gap exists between them. The evolutionary masses of these stars are closer in the two systems of tracks and are derived with a relatively higher accuracy in comparison with their ages.     

Irregular behaviour and eccentricity increase in the trajectories of a dynamical system modelling asteroidal motions

Many trajectories of the third body are integrated numerically in a modified elliptical restricted three body problem (ERTBP), in which the eccentricity, e, of the orbit of the second primary varies sinusoidally with time. It is found that, in the case of the 2:1 resonance, the introduction of the time variability of e modifies significantly the behaviour of the trajectories of the third body. In particular their osculating eccentricity e, present the following two notable features: (a) In all cases it shows a definite chaotic variation, which appears at significantly shorter time-scales than the one found by Wisdom in the e = constant case. (b) In many cases it shows a significant increase, up and beyond the (critical) value e _crit = 0.52. As a result the third body approaches the first primary at distances smaller than 0.29 (where by we denote the semi-major axis of the trajectory of the second primary around the first), which in the actual Sun-Jupiter-asteroid problem corresponds to the semi-major axis of Mars. Our result might be of interest in the context of explaining the Kirkwood gaps at the resonances where the osculating eccentricity of asteroid trajectories calculated in the classical (e = constant) ERTBP does not reach Mars crosser values.     

Decay of positronium in strong magnetic fields

We investigate the decay of bound electron-positron pairs (positronium) in strong magnetic fields (of order 10¹² Gauss, which are assumed for neutron stars) on the basis of a correct treatment of the two-body problem, thus improving previous work by Carr and Sutherland (1978). We find that, even in the presence of a strong magnetic field, the decay of the ground state of positronium must be momentum conserving, whereby the possibility of the one-photon decay is ruled out. We calculate the transition rate for the two-photon annihilation process which turns out to be larger than the field free transition rate by a factor (1/)² B/B _cr (where is the electromagnetic coupling constant, andB _cr=m _e ² c ²/(e)=4.41×10¹³ Gauss).     

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 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.     

Possibility of 24MgH+ in the solar atmosphere-high resolution rotation-vibration spectra

High resolution rotation-vibration spectra of v = 1 and 2 sequences of ²⁴MgH⁺ are obtained by a differencing process invoking the known structures of electronic bands observed in the region 214–340 nm. A reversal of R branch (band head formation) is noticed in each of the rotation-vibration bands. The J = 1 J = 0 transition is predicted to be at 376098, 365301, 354286 and 342981 MHz for v = 0,1,2 and 3, respectively, in the ground electronic state ¹⁺. The equilibrium rotational constants B _e, _e and r _e for the X ¹E⁺ state are found 6.4637cm^–1, 0.1899 cm^–1 1.6421 Å, respectively.On leave from Physics Department, Gorakhpur University, Gorakhpur, U.P., India.     

On the possibility of constructing a radiative sunspot model in magnetohydrostatic equilibrium

It is currently believed that it is impossible to construct a radiative sunspot model in magnetohydrostatic equilibrium unless magnetic fields below the surface are excessively large (> 100 kG). This belief is based on results obtained using the mixing length theory of convection. We wish to point out that by using a different theory of convection, due to Öpik (1950), it is possible to compute a radiative sunspot model in which the field becomes no greater than 9000 G. By applying two boundary conditions, (i) depth of spot equals depth of convection zone, (ii) magnetic field has zero gradient at the base of the spot, we show that a radiative spot has a unique effective temperature for a given Wilson depression, . For = 650 km, we find T _e = 3800K ; for = 150 km, T _e = 3950K. According to our model, spots having T _e cooler than these values should not exist.     

On the influence of electronic concentration fluctuations of the dispersion measure and rotation measure

The question of the influence of electronic concentration fluctuations along the line of sight on the dispersion measure (DM) and rotation measure (RM) is under consideration. it is shown that the measuring ofDM andRM enables us to determine the number of electronsN _e =n _e ds and the quantityM _e =n _e H cos ds, respectively, along the line of sight in accordance with the general assertion. Under the conditions considered, the conclusion (Lerche, 1970a, b) that the connection ofDM andRM withN _e andM _e greatly depends on the electronic concentration fluctuations, is found to be incorrect.     

Evolution of Wide Star Pairs Owing to Irregular Forces

The dynamic evolution of wide star pairs (separated by 10⁴ AU for typical galactic conditions) under the influence of random interactions with background stars during transits is studied theoretically. Transits which form a roughly equilateral triangle at some time are considered. The combined effect of many transits can be separated into systematic heating of the pair plus random diffusion over the Kepler elements a and e. The corresponding Fokker-Planck equation is derived and its range of validity indicated. A steady state distribution over a and e is obtained.     

Stability of planar oscillations of a satellite in an elliptic orbit

A problem of stability of odd 2-periodic oscillations of a satellite in the plane of an elliptic orbit of arbitrary eccentricity is considered. The motion is supposed to be only under the influence of gravitational torques.Stability of plane oscillations was investigated earlier (Zlatoustovet al., 1964) in linear approximation. In the present paper a problem of stability is solved in the non-linear mode. Terms up to the forth order inclusive are taken into consideration in expansion of Hamiltonian in a series.It is shown that necessary conditions of stability obtained in linear approximation coincide with sufficient conditions for almost all values of parameters ande (inertial characteristics of the satellite and eccentricity of the orbit). Exceptions represent either values of the parameters ,e when a problem of stability cannot be solved in a strict manner by non-linear approximation under consideration, or values of the parameters which correspond to resonances of the third and fourth orders. At the resonance of the third order oscillations are unstable, but at the resonance of the fourth order both unstability and stability of the satellite's oscillations take place depending on the values of the parameters ,e.     

Extension of the solution of Kepler's equation to high eccentricities

The classic Lagrange's expansion of the solutionE(e, M) of Kepler's equation in powers of eccentricity is extended to highly eccentric orbits, 0.6627 ... <e<1. The solutionE(e, M) is developed in powers of (e–e*), wheree* is a fixed value of the eccentricity. The coefficients of the expansion are given in terms of the derivatives of the Bessel functionsJ _n (ne). The expansion is convergent for values of the eccentricity such that |e–e*|<(e*), where the radius of convergence (e*) is a positive real number, which is calculated numerically.     

Parameter distribution of small periodic librations about the equilateral points of the elliptic restricted problem

As previously shown (Rabe, 1970), two classes of small periodic librations exist, in the plane, elliptic restricted problem, for an infinite sequence of easily specified oscillation frequenciesZ _j . The present paper considers the dependence ofZ on the eccentricitye of the primary motion, in addition to its dependence on the mass parameter , and determines the resulting relations between ande, for any given periodic frequencyZ _j . These relationships are obtained from the conditionD (Z _j ,, e)=0, where the basic determinantD has been expanded up to terms of orderZ ²⁰, ⁵, ande ⁴.Presented at the Conference on Celestial Mechanics, Oberwolfach, Germany, August 27–September 2, 1972.     

The centre-to-limb variation of the moon's brightness at 2 and 6cm wavelength

A map of the Moon at 2 cm wavelength is presented. The angular ( 1arc) and temperature resolution (< 0.1 K) is sufficient to study systematic details of the brightness distribution. In particular, the centre-to-limb variation is considered. An estimate of the dielectric constant is possible (1.4 2.5). The existence of a temperature gradient in the lunar surface layers is used to derive the depth of penetration of electromagnetic waves (L _e ), which isL _e 8 m for 2 cm wavelength. The parameters derived from the 2 cm map are found to be compatible with those obtained from a former observation at 6 cm wavelength.     

Nature of the symbiotic binary V 1329 Cyg

The published photometric and spectroscopic data of the symbiotic binary V 1329 Cyg are interpreted. It is shown, that V 1329 Cyg is an eclipsing binary with an elliptical orbit orbit (e=0.28). The cooler component fills up the Roche-lobe at periastron. A model of moving gaseous structures in the system is proposed and their influence on the radial velocity curve is shown. The following characteristics of the system are derived: the cooler component is an M6 giant with mass 7.9M , radius 339R and luminosityM _bol=–5.42, the hot component is a white dwarf surrounded by an accretion disk. The mean distance between the components is 842R and in periastron it decreases to 605R .     

Ionization equilibrium in soft X-ray-emitting solar flares

Studies of the flare-produced line feature at 1.9 Å due to highly ionized iron show that it is emitted in conditions closely approximating steady-state ionization equilibrium. Calculations of the line flux per unit emission measure from time-dependent and steady-state ionization equilibria are compared with observed values during four flares in particular. Only for electron densities N _e 10¹⁰ cm^-3 do the time-dependent equilibrium values give as good an approximation to the observed values as the steady-state equilibrium. This lower limit is compared with values of N _e derived from analyses of the temperature decline in each of these events, and with estimates of N _e given by other workers.NAS/NRC Research Associate.     

A strong limitation on the rapidity of flux-pile-up reconnection

The reconnection rate which can be achieved in the steady-state flux-pile-up regime is severely limited by the gas pressure of the plasma. Using the family of solutions obtained previously by Priest and Forbes, we show that the Alfvén Mach number of the plasma flowing towards the reconnection site cannot exceed _e/(8 ln R _me), where _e and R _me are the plasma and magnetic Reynolds numbers at large distance. This limit corresponds to a very weak flux-pile-up, and it is a factor of _e slower than the maximum Petschek rate. Thus the maximum flux-pile-up reconnection rate in the corona is at least two orders of magnitude smaller than the rate implied by flare observations.     

A general algorithm for the solution of Kepler's equation for elliptic orbits

An efficient algorithm is presented for the solution of Kepler's equationf(E)=E–M–e sinE=0, wheree is the eccentricity,M the mean anomaly andE the eccentric anomaly. This algorithm is based on simple initial approximations that are cubics inM, and an iterative scheme that is a slight generalization of the Newton-Raphson method. Extensive testing of this algorithm has been performed on the UNIVAC 1108 computer. Solutions for 20 000 pairs of values ofe andM show that for single precision (10^–8) 42.0% of the cases require one iteration, 57.8% two and 0.2% three. For double precision (10^–18) one additional iteration is required. Single- and double-precision FORTRAN subroutines are available from the author.     

High resolution EUV structure of the chromosphere-corona transition region above a sunspot

Ion emission line intensities between 1170 and 1700 Å allow one to determine the differential emission measure (DEM) and electron pressure of the plasma in the solar transition region (TR). These line intensities together with their Doppler shifts and line widths are measured simultaneously for the first time above a sunsport from data obtained with the NRL High Resolution Telescope and Spectrograph with 0.06 Å spectral and 1 spatial resolution.The Doppler shifts show both subsonic and supersonic flow in the same line of sight over the umbra. The temperature structure for 40 resolution elements in the sunspot umbra and penumbra is derived from the DEM and the observed electron pressures.Extrapolation of the emission measure curves supports the previous EUV and X-ray observations that coronal plasma above sunspots with T _e>10⁶ K is reduced while emission from TR plasma between 2×10⁵ and 10⁶ K is greatly enhanced relative to quiet or active regions. This enhancement shifts the minimum of the DEM to lower temperatures and increases the slope at 2×10⁵ K by a factor of two.New pressure diagnostics using the emission line intensity ratios of C iv to N iv are presented, and applied to the data.The energy balance in the TR for the sunspot umbra is dominated by radiative losses from the large amount of TR plasma.An estimate of the energy budget shows that an energy input is required to balance the radiative energy losses above the umbra. The observed divergence of the enthalpy flux for the umbral downflows can balance these radiative losses for T _e between 30000 and 200 000 K.A typical umbral model of T _e versus reduced mass column density is compared with one for chromospheric temperatures determined from the Ca H and K lines.Institute of Theoretical Astrophysics, University of Oslo, Norway.     

The physical conditions in inner corona derived from spectral data of the solar eclipse on 7 March, 1970

The spectrum of the solar corona obtained during the eclipse of 7 March, 1970 has been studied. From the analysis of the line halfwidths it is concluded that the emission of coronal lines of various groups arises in different volumes of the corona. The lines 5303 and 4231 are analyzed in detail and by the use of the equations of excitation and ionization balance. The distribution of electron temperature T _e and density n _e vs height in the solar corona has been found. The usually adopted n _e and T _e are noticeably less than those values obtained in present analysis. This discrepancy may be due to the fact that the considered area of the corona seems to be in an unusual condition.