Fourier parameters and moments of polarization profiles of magnetically active lines. Fourier vector magnetograph

A new method is proposed to determine all components of the solar magnetic fields using the cumulants of the profile of a magnetic sensitive line. The method is based on polarization measurements in a number of points of the line profile and subsequent calculation of the amplitudes and phases of its two first Fourier-harmonics.     

Models, figures and gravitational moments of Jupiter’s satellite Io: Effects of the second order approximation

The effects of the equilibrium figure theory to within terms of the second order in a small parameter α on figure parameters and gravitational moments of the Galilean satellite Io have been considered. Integro-differential equations of the theory of figure to second order have been first solved numerically. Relations between the low-order coefficients of the gravitational field for satellites in hydrostatic equilibrium are generalized according to the second order theory. To show the effects of the second approximation, two three-layer trial models of Io are used. The considered models of the Io’s interiors differ by the size and density of the core, while having the same thickness and density of the crust, and the mantle density difference is only 20 kg/m³. The corrections of second order in smallness to the gravitational moments J₂ and C₂₂ decrease the third decimal digit of model gravitational moments by two units. As the effects of third and forth harmonics are determined mostly by outer layers of Io, to distinguish between model mantle density, the gravitational moments J₄, C₄₂ and C₄₄ should be determined to accuracy with three or four decimal digits. The second order corrections mostly effect the semi-axis a, and less the semi-axes b and c.     

A first order analysis of variations of the limb darkening and the shapes for solar Fraunhofer lines

New center-to-limb measurements in FeI lines show changes in both the line profiles and the limb darkening curves that appear to be characteristic of many other solar lines. Here we seek the constraints placed on the atmospheric model by these effects. We find that in addition to a depth varying source function we must also allow the ratio of the continuous absorption coefficient to the total absorption coefficient to pass through a minimum in the mid-photosphere. Such an effect is consistent with inward increases of the Doppler width and damping constant in the upper photosphere and an inward increase of the ionization for both iron and hydrogen in the low photosphere.The National Center for Atmospheric Research and Kitt Peak National Observatory are sponsored by the National Science Foundation.     

Determination of mass-loss rates from the first order momentW 1 of unsaturated P Cygni line profiles

The escape probability method introduced by Sobolev to treat the transfer of line photons is used in order to derive the expressions of thenth order momentW _n (E()/E _c –1)(–₁₂) ⁿ ·d of a P Cygni profile formed in rapidly expanding envelopes around a central point-like source under various physical and geometrical conditions.With the only assumption that there is mass-conservation of the species in the flow, we state for the case of optically thin lines that the relation between the first order momentW ₁ and the quantityM _n (level), first established by Castoret al. (1981) under more restrictive conditions, is in fact independent of the type of velocity fieldv(r) and a fortiori of the distribution adopted for the radial opacity ₁₂ ^r (X). These results also remain unchanged when including collisions (0) and/or an additional rotational velocity fieldv (r) in the expanding atmosphere. We investigate the presence of an underlying photospheric absorption line and conclude that for realistic cases, neglecting this boundary condition to the radiative transfer leads to an underestimate of the mass-loss rate by a factor of about 20%. By means of a three-level atom model, we demonstrate that all results derived for a single line transition equally apply for an unresolved doublet profile providedW ₁ andM _n (level) are calculated with the weighted wavelength _D and total oscillator strengthf _D of the doublet.Considering the occultation and inclination effects caused by the finite size of the central core, we refine the value of the multiplicative constant fixing the ratio ofW ₁ toM _n (level). We show that this relation allows a determination of the mass-loss rate with an uncertainly less than 30%, irrespective of the sizeL _max of the atmosphere and of the limb-darkening affecting the stellar core. Reviewing all possible sources of error, we finally conclude that this method of deriving a mass-loss rate from the analysis of an unsaturated P Cygni profile is very powerful. The total uncertainty affecting the determination ofM _n (level) from the measurement ofW ₁ should be smaller than 60%.     

A generalized method for deriving mass-loss rates: The first order moment of unsaturated P Cygni line profiles

For the case of optically thin lines, we show that the relation existing between the first order momentW ₁ (E()/E _c –1)(–₁₂)d of a P Cygni profile and the quantityMn(level), whereM is the mass-loss rate of the central star andn(level) the fractional abundance of the ion under study, is in fact independent of any Sobolev-type approximations used for the transfer of line radiation, contrary to what has been assumed in some previous works (Castoret al., 1981; Surdej, 1982). Consequently, all results established in the context of very rapidly expanding atmospheres and mainly referring to the non-dependence ofW ₁ vs various physical (underlying photospheric absorption line, limb darkening, etc.) and geometrical (velocity fieldv(r), etc.) effects remain unchanged for arbitrary (e.g non-Sobolev type) outward-accelerating velocity laws.Whenever applied with caution, Equation (50) thus provides a very powerful means of deriving mass-loss rates—with a total uncertainty less than 60 percent—from the measurementW ₁ of unsaturated P Cygni profiles observed in the spectrum of early—as well as late—type stars, quasars, etc.     

The profiles of fraunhofer lines in the presence of zeeman splitting

For the case of pure absorption lines (LTE) a method is described which enables the general computation of Zeeman-split line profiles. The magnetic field vector, the Doppler shift and the line absorption coefficient is permitted to vary arbitrarily with optical depth. Elliptical birefringence (e.g., Faraday rotation) of the solar atmosphere is taken into account. Some numerical examples are given and some interesting behaviors of the line profiles are discussed.     

The hydrogen balmer lines and the structure of the quiet solar chromosphere

Spectra of Hα, Hβ and Hδ have been taken under good seeing conditions with the vacuum tower telescope of Sacramento Peak Observatory. Intensity curves are given at various wavelengths in these lines to permit further comparison with a theoretical model. Moreover, considering in each case the range of height in which the lines are almost optically thin and using a few approximations, the following results are derived: between 2000 and 6000 km above the limb the average thermal + turbulent velocity of the atoms is found to increase from 20 km s^?1 to 30 km s^?1 and the mean number of hydrogen atoms per cm³ in level 2 is given by $$\log n_2 {\text{ = }}4.5{\text{ }} - {\text{ 0}}{\text{.00056(}}z - 2000)$$ z being the altitude above the limb in km. For line profile computations a new interpolation formula is presented; it gives good profiles with a small number of scans, saving microphotometer time.     

Intensity and polarization line profiles in a semi-infinite Rayleigh-scattering planetary atmosphere. III. Variation of polarization profiles and the Stokes parameter Q over the disk

The variation of the polarization profiles, the Stokes parameters Q andU, and the angle defining the plane of polarization along the intensity equator and along the mirror meridian, on whichμ = μ ₀, in a Rayleighscattering atmosphere is studied. It is found that these variations are more complex than thought hitherto, particularly at large phase angles.     

The behaviour of the infrared lines of neutral oxygen and the balmer lines of hydrogen in the spectra of early-type stars with emission lines

The equivalent widths of the oxygen lines at 7774 and 8446 and of H (and some H) have been measured for 22 early-type, emission-line stars. A strong correlation between H and 8446 intensities has been found, although there is no such correlation between H and 7774. This confirms the probability that Bowen's mechanism is operative (the neutral oxygen 3³ D state is overpopulated because the excitation energy of Ly- nearly coincides with that of theOi 1025 line). The possibility of using 8446 and H equivalent widths for a comparison of oxygen to hydrogen abundances in these stars is discussed.     

The theory of the nutation for the rigid earth model at the second order

We perform a complete reconstruction of the series of the nutation for a rigid Earth model with the use of the very accurate theories ELP2000 and VSOP82 for the motion of the Moon and the planets respectively, in such a way that all the individual contributions up to 0.005 mas should be taken. This implies the introduction of the planetary effects, of the influence of second-order parts of the potential of the Earth (J3, triaxiality), and some improvements due to an extension of the theory at the second order. All this increase notably the number of coefficients to be taken in account, and modifies also in a significant way the value of some of them.     

CMB recombination lines of hydrogen: The differential spectrum

We have calculated the distortions of the cosmic microwave background (CMB) spectrum in the wavelength range 2–50 cm due to the superposition of the CMB hydrogen recombination radiation in subordinate lines. The level populations were determined by numerically solving the equation of recombination kinetics together with the statistical equilibrium equations for a 60-level model hydrogen atom. The relative distortions are ≈10^?7–10^?6, with their wavelength dependence having a low-contrast, wavy pattern. However, the contrast increases severalfold and becomes pronounced when passing to the differential distortion spectrum. We study the dependence of the distortions on cosmological parameters.     

Theoretical study of the Fraunhofer lines polarization: The case of Ca i 4227

The measurements by Brückner (1963) of the Ca i 4227 polarization at the Sun's limb provides us with a test for the theory of line polarization. Computations are developed taking into account: (a) the transfer polarization, due to the anisotropy of radiation field; (b) the depolarizing collisions acting in the wings. The magnetic field is not taken into account and the theory is not valid in the Doppler core. In the wings a very good fit is obtained, using appropriate source-functions fitting the observed profiles at the center of the disk, and from center to limb.     

The effect of angle-dependent non-coherent electron scattering and partial redistribution on the polarization of resonance lines

The polarized radiative transfer equation is solved when angle-dependent partial redistribution and non-coherent electron scattering are included as line-scattering mechanisms. A static atmosphere with plane parallel symmetry is assumed. Test calculations are used to illustrate the effects of the electron-scattering coefficient, the thermalization parameter and the continuous absorption coefficient on the line polarization. Results of angle-averaged and angle-dependent redistribution functions are compared and it is shown that angle-dependent functions should be used to model the wing polarization of optically thin lines. The lower the continuous absorption compared with the electron scattering, the higher the wing polarization.     

First and second order approximations of the first adiabatic invariant for a charged particle interacting with a linearly polarized hydromagnetic plane wave

In the present paper the effect of a sinusoidal modulation of an electromagnetic field on the invariance of the magnetic moment is studied. Such a generalized invariant plays an important role in problems concerning the motion of charged particles in the non-uniform magnetic field of the magnetosphere or the solar wind. In order to find an adiabatic invariant J, a canonical transformation is introduced, and J is expanded in an asymptotic series in the relative modulation amplitude. We are studying the first and second order terms of this expansion. It is further shown that the curves J = constant closely fit the results obtained by a numerical integration of the system of differential equations governing the motion of the particles.     

The role of non-thermal electrons in the hydrogen and calcium lines of stellar flares

There is observational evidence showing that stellar and solar flares occur with a similar circumstance, although the former are usually much more energetic. It is expected that the bombardment by high-energy electrons is one of the chief heating processes of the flaring atmosphere. In this paper we study how a precipitating electron beam can influence the line profiles of Ly α , H α , Ca  ii K and λ 8542. We use a model atmosphere of a dMe star and make non-LTE computations taking into account the non-thermal collisional rates owing to the electron beam. The results show that the four lines can be enhanced to different extents. The relative enhancement increases with increasing formation height of the lines. Varying the energy flux of the electron beam has different effects on the four lines. The wings of Ly α and H α become increasingly broad with the beam flux; change of the Ca  ii K and λ 8542 lines, however, is most significant in the line centre. Varying the electron energy (i.e. the low-energy cut-off for a power-law beam) has a great influence on the Ly α line, but little on the H α and Ca  ii lines. An electron beam of higher energy precipitates deeper, thus producing less enhancement of the Ly α line. The Ly α /H α flux ratio is thus sensitive to the electron energy.     

On the nonthermal excitation and polarization of X-ray lines during solar flares

The energy distributions of nonthermal electrons are derived from hard X-ray spectra taken during the impulsive phase of two 2B flares in February 1969. They are used to calculate the fluxes of nonthermally excited X-ray lines of hydrogen-like and helium-like ions. These fluxes are compared to the total line fluxes observed at the same time with crystal spectrometers. The nonthermal excitation is found to give only small contributions to the total line intensities. This implies that the impact polarization which is to be expected for anisotropic velocity distributions of the energetic electrons, will be low. Nevertheless it should be feasible to detect line polarization during the impulsive phase of strong X-ray flares.NAS/NRC Research Associate.     

Radial matrix elements of hydrogen atom and the correspondence principle

Radial dipole matrix elements having astrophysical importance have been computed for highly excited states of hydrogen atom. Computation is based on Heisenberg’s form of correspondence principle for Coulomb potential. Particular attention has been paid to the choice of classical analogue(n _c) of principal quantum number (n). The computed radial matrix elements are in good agreement with quantum mechanical results. Further, radial matrix elements for few transitions involving highn neighboring states of hydrogen atom are presented.