Sobolev's Φ-function of radiative transfer in planar and spherical scattering media

In recent years Sobolev's -function of radiative transfer has been discussed in connection with the resolvent of Milne's integral equation such that it plays an important role in the determination of the radiation field in semi-infinite (or finite) atmospheres with internal sources (cf. Sobolev, 1963). In the present paper, the part of Sobolev's -function in plane-parallel and spherical, isotropically scattering, atmospheres with internal source distribution is investigated from analytical and numerical aspects. With the aid of invariant imbedding (cf. Bellmanet al., 1968), we computed Sobolev's -function of Milne's integral equation for the planar case by solving the Cauchy system for the auxiliary function and Chandrasekhar'sX- andY-functions. The corresponding -function for the spherical case is readily obtained from the for the planar case.Investigation supported by the National Science Foundation under Grant No. GP 29049, the Atomic Energy Commission, Division of Research under Contract No. AT(04-3)-113, Project 19, and the National Institutes of Health under Grant No. 16197-05.     

A possible interpretation of the optical spectrum for BL Lac-type objects

On the basis of Sobolev's method, the population of 30 levels of hydrogen atom is determined allowing for the radiative and collision processes of the heating and ionization of the medium with velocity gradient gradv=10^–9–10^–11s^–1, electron temperatureT _e=10⁴ K-2×10⁴ K and electron densityN _e=10¹⁰ cm^–3–10¹¹ cm^–3. The central source radiation is characterized by a power spectrum with spectral indices varying from 0 to 2. A region of possible physical conditions is found where the thermal diffuse radiation of the envelope exceeds the emission in the Balmer H line.     

Analysis of the spectral features of a recent supernova 1987n

We present an attempt to analyse the spectra of SN 1987n in NGC 7606, covering a period of 10 days from the time of maximum brightness. The velocities in the rest frame of NGC 7606 and the depths of the spectral lines at maximum light are very close to those of SN 1981b in NGC 4536, slight differences being nevertheless present. A distance to NGC 7606 of 46±11 Mpc for anH ₀ value of 50 km s^–1 Mpc^–1 is inferred from Pskovskii's relation.The analysis carried out within a standard simple model of spectral synthesis involves a high uncertainty in the abundance determinations for the intermediate-mass elements quoted. This uncertainty arises on the one hand from the free choice of the excitation temperature and from the sensitivity to changes in the excitation temperature of the depths of the strongest lines of those elements and, on the other hand, from the impossibility of obtaining within this model an estimate of other abundances — He, O, Na, S — which have NLTE populations.The analysis developed in a new model based (as is the standard one) on Sobolev's approximation but allowing for a more realistic continuum treatment points to an important attenuation effect on the radiation in the lines, due to the continuum scattering, which can also affect abundance determinations.     

X-ray nova candidates among old classical novae

A review of light curves of known x-ray novae made it possible to identify criteria by which x-ray nova candidates were selected among old novae: amplitude of optical outburst 7–10 ^m, shape of light curve during the outburst with a temporary fading by 2–3^m lasting up to four days and an abrupt final fading from the 6^m level (relative to the quiet state). We identified LS And, AL Com, V592 Her, and HV Vir as x-ray nova candidates. Recurrent outbursts should be expected for the first and third stars. Less reliable candidates are V341 Nor, V787 Sgr, and V719 Sco. A possible recurrent nova candidate may be V1330 Cyg. Translated from Astrofizika, Vol. 42, No. 3, pp. 359–364, July–September, 1999.     

Solution of the transfer equation for interlocked multiplets by probabilistic method

Sobolev's probabilistic method — The method of quantum exit from the medium — has been applied to solve the transfer equation for the case of interlocking without redistribution. The solution contains the function (x) which is same as theH-function involved in the solution given by Busbridge and Stibbs the method of principle of invariance.     

Rayleigh scattering: Volterra equation for the matrix source function

Multiple Rayleigh scattering is examined in a semi-infinite atmosphere with uniformly distributed primary sources of partially polarized radiation. The resulting linear polarization is described by a 2×2 matrix transfer equation. A matrix generalization of Rybicki's two point Q-integral is obtained for this case. It is shown that the Volterra equation for the matrix source function for this problem is a particular case of our Q integral. Applying the Laplace transform to it yields the matrix form of the Ambartsumyan-Chandrasekhar H-equation. The Volterra equation for Sobolev's matrix resolvent function is another simple consequence of this equation. Translated from Astrofizika, Vol. 52, No. 2, pp. 301–310 (May 2009).     

An efficient method to calculate Ambarzumian-Chandrasekhar's and hopf's functions

An efficient method is proposed to calculate scalar Ambarzumian-Chandrasekhar's and Hopf's functions. This method is based on the approximation of Sobolev's resolvent function using exponent series, the coefficients of which are readily found from approximate characteristic equation and from a system of linear algebraic equations.The approximate expressions for the above functions are given. For checking purposes the calculations were carried out in single, double, and quadruple precision. For isotropic, Rocard, and Rayleigh scattering we present a sample of results in 14 significant figures.The Hopf function for isotropic and Rayleigh scattering is presented in 18 significant figures and the well-known Hopf constantq() is found in 59 significant figures.     

A novel methodology for radiative transfer in a planetary atmosphere

A novel methodology for evaluating the field of anisotropically scattered radiation within a homogeneous slab atmosphere of arbitrary optical thickness is provided. It departs from the traditional radiative transfer approach in first considering that the atmosphere is illuminated by an isotropic light source. From the solution of this problem, it subsequently proceeds to that for the more conventional case of monodirectional illumination. The azimuthal dependence of the field is separated in the usual manner by an harmonic expansion, leaving a problem in four dimensions (=optical depth, ₀=thickness, , =directions of incidence and scattering) which, as is well known, is numerically extremely inconvenient. Two auxiliary radiative transfer formulations of increasing dimensionality are considered: (i) a transfer equation for the newly introduced functionb ^m(,,₀) with Sobolev's function ^m(,₀) playing the role of a source-function. Because the incident direction does not intervene, ^m is simply expressed as a single integral term involvingb ^m. For bottom illumination, an analogous equation holds for the other new functionh ^m(,,₀). However, simple reciprocity relations link the two functions so that it is only necessary to considerb ^m; (ii) a transfer equation for the other new functiona ^m(,,,₀) with a source-function provided by Sobolev's functionD ^m(,,₀). For bottom illumination, another functionf ^m(,,,₀) is introduced; by a similar argument using reciprocity relations,f ^m is reduced toa ^m rendering necessary only the consideration ofa ^m. However, a fundamental decomposition formula is obtained which shows thata ^m is expressible algebraically in terms of functions of a single angular variable. The functions ^m andD ^m are shown to be the values in the horizontal plane ofb ^m anda ^m, respectively. The other auxiliary functionsX ^m andY ^m are also expressed algebraically in terms ofb ^m. These results enable one to proceed to the final step of evaluating the radiation field for monodirectional illumination. The above reductions toalgebraic relations involving only the functionb ^m appear to be more advantageous than Sobolev's (1972) recent approach; they also circumvent some basic numerical difficulties in it. We believe the present approach may likewise prove to be superior to most (if not all) other methods of solution known heretofore.This paper presents the results of one phase of research carried out at the Jet Propulsion Laboratory under Contract No. NAS-7-100 sponsored by the National Aeronautics and Space Administration.     

Line transfer in supernovae envelopes by Sobolev's treatment

We combine an accurate treatment of continuum transfer in supernovae envelopes with a line transfer treatment using Sobolev's approximation to analyze the influence of some physical parameters (the continuum opacity, the ratio of scattering over absorption and the temperature profile) on the emergent intensity of a resonant scattering line.Paper presented at the 11th European Regional Astronomical Meetings of the IAU on New Windows to the Universe, held 3–8 July, 1989, Tenerife, Canary Islands, Spain.     

V4334 Sgr (Sakurai's Object): The Distance Problem

The central star V4334 Sgr (Sakurai's Object) of the planetary nebula PN G010.4+04.4 underwent in 1995–1996 the rare event of a very late helium flash. It represents only one out of two such events during the era of modern astronomy (the other event was V605 Aql = Nova Aql 1919). All the other prominent objects of that type originate from events occurring several thousands of years ago (e.g. A30, A78). Thus it is of special interest for stellar evolution theory to model the detailed observations obtained during the last four years. Those models depend essentially on basic stellar parameters like effective temperature, surface gravity and stellar radius. Most of them depend strongly on the assumed distance to the object. Some models may give some constraints on this parameter, but most of them depend on the assumption as input parameter. Hence to determine a reliable distance is of considerable significance. This should be obtained through models that give us lower and upper boundaries, or through means which are independent of models. The detailed review, by using every kind of determination available up to now, leads to a Galactic foreground extinction of E _B–V =0^m75 ±0.05 and a distance of D = 2.0_-0.6 ^+1.0 kpc.     

Photometry of the contact binary system V471 Cas

UBV photoelectric observations of the W UMa-type binary V471 Cas were made from September to November 1984. Its colour indices were found to be(B-V)=0 _. ^m 771±0 _. ^m 02 and(U-B)=0 _. ^m 196±0 _. ^m 02. TheU, B, andV light curves of V471 Cas show some photometric fluctuations. We found that its orbital period is not 0.335998 days which was given by GCVS (1986), but 0.405356 days.Photometric orbital elements of V471 Cas were found using the Wilson-Devinney method. V471 Cas is a contact system, in which the overcontact factor is 0.19, its mass ratio of two components is 0.5947, and orbital minclination is in 83.29 degrees.     

An observational study of the high-amplitude δ Scuti star V367 Cam

CCD observations of V367 Cam, a high-amplitude δ Scuti star, were carried out in both V and R band on October 22 and 23 and November 4, 2016. By using the new observed maximum times and the previously reported epochs, a new ephemeris is obtained. Fourier analysis of the light curves yields the main frequency f1 = 8.2242 d ⁻¹ and its second harmonic as 2f1 = 16.4062 d ⁻¹. The pulsation constant specifying the oscillation characteristics of pulsating stars is calculated from physical parameters of V367 Cam. We obtained Q = 0.03814 ± 0.00409, which demonstrates this star pulsates in single radial fundamental mode. With the derived physical parameters, the evolution status of V367 Cam is determined and also it is shown on H-R diagram against other δ Scuti stars. Furthermore, we made a count of mono-periodic HADS, which are expected to be decreasing in number and turned to be multiperiodic in the advent of high precision photometry and extended data.     

Photometric Analysis and Period Investigation of the EW Type Eclipsing Binary V441 Lac

Four color light curves of the EW type eclipsing binary V441 Lac were presented and analyzed by the W–D code. It is found that V441 Lac is an extremely low mass ratio (q = 0.093±0.001) semi-detached binary with the less massive secondary component filling the inner Roche lobe. Two dark spots on the primary component were introduced to explain the asymmetric light curves. By analyzing all times of light minimum, we determined that the orbital period of V441 Lac is continuously increasing at a rate of dP/dt = 5.874(±0.007) × 10^?7 d yr^?1. The semi-detached Algol type configuration of V441 Lac is possibly formed by a contact configuration destroyed shallow contact binary due to mass transfer from the less massive component to the more massive one predicted by the thermal relaxation oscillation theory.     

UBV photometry of Vesta

A lightcurve of Vesta, obtained on four nights between June 23 and 30 June 1978 during the coordinated campaign for the determination of the rotation period, is presented. The observations were performed at the 91-cm telescope of the Catania Observatory employing UBV filters and a photon counting photometer. The V lightcurve apparently shows a single maximum, suggesting that the 5^h20^m29^s.2 period is the correct one. Features are evident near the maximum and the minimum closely resembling those of Johnson's lightcurve of 22 December 1950 and Taylor's of January 21, 1973. The amplitude in V light is 0^m.105 and small variations are also found in the color indices. The largest color variation is for the U-V with Δm = 0.^m.05, which is slightly larger than the value 0^m.02 found by T. Gehrels [Astron. J.72, 929 (1967)]. The maximum and minimum values occur at the same phase with respect to the maximum V light as found by Gehrels, i.e., Vesta appears bluer near 0^p.25 and redder near 0^p.7. Corrections with the solar phase angle were made using the coefficients given by Gehrels for the B-V and U-V while a new value of 0.036 mag/deg was assumed for the V observations. The available amplitudes of Vesta's lightcurve were analyzed with respect to the longitude position and the solar phase angle.     

Photometric study of the FUor star V 1735 Cyg (Elias 1-12)

Results from optical photometric observations of the PMS star V 1735 Cyg are reported. The star is located in the IC 5146 dark cloud complex—a region of active star formation. On the basis of observed outburst and spectral properties, V 1735 Cyg was classified as a FUor object. We present data from BVRI CCD photometric observations of the star, collected from March 2003 to January 2009. Plates from the Rozhen Schmidt telescope archive were scanned for a brightness estimation of the star. A sequence of sixteen comparison stars in the field of V 1735 Cyg was calibrated in BVRI bands. The data from photographic observations made from 1986 to 1992 show a strong light variability (ΔV=1_⋅^m2). In contrast, the recent photometric data obtained from 2003 to 2009 show only small amplitude variations (ΔI=0_⋅^m3). The analysis of existing photometric data shows a very slow decrease in star brightness—1_⋅^m8 (R) for a 44 year period. The possibilities for future photometric investigations of V 1735 Cyg using the photographical plate archives is discussed briefly.     

Energy spectra of the main groups of galactic cosmic rays in the model of three classes of sources

We suggest a model to consistently describe the available experimental data on the elemental cosmic-ray energy spectra obtained in direct measurements and to make a smooth transition to the spectrum of all particles measured with extensive air showers. The model suggests the existence of three classes of cosmic-ray sources—shocks from supernova explosions that produce power-law rigidity spectra with different maximum rigidities and different spectral indices. The shocks from high-mass supernovae exploding in OB associations are assumed to be the most powerful class of sources. This class of sources accelerates cosmic rays to a maximum rigidity of 4 × 10¹⁵ V. The shocks from nonassociated supernovae exploding into a random interstellar medium are assumed to be the next class (in order of decreasing power). This class of sources accelerates cosmic rays to a maximum rigidity of 5 × 10¹³ V. The third, weakest class of sources is assumed to accelerate cosmic rays to a maximum rigidity of 2 × 10¹¹ V. Nova explosions could be possible physical objects in this class.     

Analysis of plasma and field conditions during some intensely geo-effective transient solar/interplanetary disturbances of solar cycle 23

The problem of solar wind-magnetosphere coupling is investigated for intense geomagnetic storms (Dst < -100nT) that occurred during solar cycle 23. For this purpose interplanetary plasma and field data during some intensely geo-effective transient solar/interplanetary disturbances have been analysed. A geomagnetic index that represents the intensity of planetary magnetic activity at subauroral latitude and the other that measures the ring current magnetic field, together with solar plasma and field parameters (V, B, Bz, σB, N, and T) and their various derivatives (BV,-BVz, BV², -BzV², B²V, Bz²V, NV²) have been analysed in an attempt to study mechanism and the cause of geo-effectiveness of interplanetary manifestations of transient solar events. Several functions of solar wind plasma and field parameters are tested for their ability to predict the magnitude of geomagnetic storm.     

Valence state partitioning of V between pyroxene‐melt: Effects of pyroxene and melt composition,and direct determination of V valence states by XANES. Application to Martian basalt QUE 94201 composition

Abstract— Experiments on a Martian basalt composition show that D_v augite/melt is greater than D_v pigeonite/melt in samples equilibrated under the same fO₂ conditions. This increase is due to the increased availability of elements for coupled substitution with the V³⁺ or V⁴⁺ ions, namely A1 and Na. For this bulk composition, both A1 and Na are higher in concentration in augite compared with pigeonite; therefore more V can enter augite than pigeonite. Direct valence state determination by XANES shows that the V³⁺ and V⁴⁺ are the main V species in the melt at fO₂ conditions of IW‐1 to IW+3.5, whereas pyroxene grains at IW‐1, IW, and IW+1 contain mostly V³⁺. This confirms the idea that V³⁺ is more compatible in pyroxene than V⁴⁺. The XANES data also indicates that a small percentage of V²⁺ may exist in melt and pyroxene at IW‐1. The similar valence of V in glass and pyroxene at IW‐1 suggests that V²⁺ and V³⁺ may have similar compatibilities in pyroxene.     

Paschen-line Stark-broadening as an electric field diagnostic in erupting prominences

We analyse the Stark broadening of the Balmer and Paschen lines emitted in two bright eruptive prominences, to determine the total electric field in these structures. We show that the Paschen lines provide a significantly more sensitive and accurate electric field indicator than the Balmer lines used previously in such studies. In the two eruptive events analysed here, the total electric fields agree to within 5–10 V cm^-1 with the pressure-broadening fields expected from local densities of the cool plasma, measured simultaneously and co-spatially by a line-ratio diagnostic. We conclude that in such structures the upper limit to any widespread macroscopic fields is roughly 10 V cm^-1 or less. This is in agreement with the motional electric field that might be associated with reconnection at the observed rate of the prominences' outward motion of about 135 km s^-1.     

Optical photometry of V4334 Sgr (Sakurai's Object)

The optical photometric evolution of the final helium flash object V4334 Sgr from 1994 to 2001 is described. The rise to optical maximum (1994–1996) is characterized by a continuous increase of colour indices, indicating a slowly expanding, cooling pseudo-photosphere. This photosphere became stationary in 1997. In the following years, the object underwent brightnessdeclines of increasing strength, which are similar in character to the `reddeclines' of RCB stars. The fading of V4334 Sgr is more dramatic than anybrightness decline of an RCB star: at present, only 10^-5 of the visual light reaches the observer. Most short-wavelength radiation is absorbed by a dust envelopethat completely surrounds the star, and is re-radiated in the infrared. The sparse optical data of 2000–2001 show that the obscuration has not increased in strength any more. The light curve of V4334 Sgr is similar to that of the final helium flash object V605 Aql which erupted in 1919.