The polarization-free approximation

The development of effective iterative methods capable of accurately solving NLTE Stokes transfer problems is of considerable importance for the investigation of solar and stellar magnetic fields. After briefly indicating the iterative approach which is being presently pursued for the exact solution of such problems, the particular regime where polarization signals can only be due to the Zeeman effect is considered in some detail. By means of NLTE Stokes transfer calculations for a two-level atomic model it is first shown that the currently-used field-free approximation (Rees, 1969) cannot be safely applied in the presence of magnetic field gradients. Such gradients lead to changes in the shape and width of the line profiles and they can produce non-negligible effects on the atomic level populations and line source functions. A new approximate method is then proposed, which does not require the actual solution of the Stokes vector transfer equation and is practically as fast as the field-free one. This polarization-free approximation provides a fairly good account of the effects of homogeneous and inhomogeneous magnetic fields on the statistical equilibrium and is very easy to implement in any existing non-magnetic, multi-level transfer code.On leave from the Dipartimento di Astronomia e Scienza dello Spazio, Università di Firenze, Largo E. Fermi 5, I-50125 Firenze, Italia     

A numerical solution of MHD free-convection flow in the general nonlinear stokes problem by the finite-difference method

With viscous dissipation and Joule heating taking into account a numerical solution of magnetohydrodynamic free convection flow, in the Stokes's problem, is obtained for different values of Prandtl numberP. The fluid is viscous, incompressible, and electrically conducting and the magnetic lines of force are assumed to be fixed relative to the plate which is started moving impulsively in its own plane (ISP) or it is uniformly accelerated (UAP). The solution is obtained with an implicit second-order method, forP=0.71 (air) andP=7 (water) and the obtained results are shown on figures and tables.     

The first and second order moments of the polarization profiles of hydrogen lines

The main properties of the first- and second-order moments of polarized hydrogen lines, forming in the presence of stationary electric and magnetic fields, are reviewed. The analytical results presented here apply directly to the case of optically-thin emission lines in the LTE regime. Some applications of such results to electric- and magnetic-field diagnostics in (solar) plasmas are then briefly considered.On leave from the Dipartimento di Astronomia e Scienza dello Spazio, Università di Firenze, Largo E. Fermi 5, I-50125 Firenze, Italy     

Numerical Integration of Regular Equations of the Planar Motion of Terrestrial Bodies

The parameters of L matrices are applied to the numerical integration of regular equations describing the motion of minor bodies in the Solar System. The problem of the optimal choice of the regularizing change of variables is formulated in the context of the numerical integration of the equations of motion using the Runge–Kutta–Fehlberg method. Arbitrary perturbations are taken into account. This problem is completely solved in the case of planar motion. The solution of the optimization problem reduces the amount of computations needed to determine the vector of perturbing accelerations. Results of numerical integrations are given.     

Interplanetary magnetic field structure and solar wind parameters as inferred from solar magnetic field observations and by using a numerical 2-D MHD model

An attempt is made to infer parameters of the solar corona and the solar wind by means of a numerical, self-consistent MHD simulation. Boundary conditions for the magnetic field are given from the observations of the large-scale magnetic field at the Sun. A two-region, planar (the ecliptic plane is assumed) model for the solar wind flow is considered. Region I of transonic flow is assumed to cover the distances from the solar surface up to 10R _S (R _S is the radius of the Sun). Region II of supersonic, super-Alfvénic flow extends between 10R _S and the Earth's orbit. Treatment for region I is that for a mixed initial-boundary value problem. The solution procedure is similar to that discussed by Endler (1971) and Steinolfson, Suess, and Wu (1982): a steady-state solution is sought as a relaxation to the dynamic equilibrium of an initial state. To obtain a solution to the initial value problem in region II with the initial distribution of dependent variables at 10R _S (deduced from the solution for region I), a numerical scheme similar to that used by Pizzo (1978, 1982) is applied. Solar rotation is taken into account for region II; hence, the interaction between fast and slow solar wind streams is self-consistently treated. As a test example for the proposed formulation and numerical technique, a solution for the problem similar to that discussed by Steinolfson, Suess, and Wu (1982) is obtained. To demonstrate the applicability of our scheme to experimental data, solar magnetic field observations at Stanford University for Carrington rotation 1682 are used to prescribe boundary conditions for the magnetic field at the solar surface. The steady-state solution appropriate for the given boundary conditions was obtained for region I and then traced to the Earth's orbit through region II. We compare the calculated and spacecraft-observed solar wind velocity, radial magnetic field, and number density and find that general trends during the solar rotation are reproduced fairly well although the magnitudes of the density in comparison are vastly different.     

Radiative transfer in spectral lines by noncoherent scattering

The functional analytic method of solution is applied to investigation of the radiative transfer equation in spectral lines. A problem of scattering in the spectral line with the frequency redistribution in anisotropic-scattering infinite and semi-infinite media is considered. Continuum absorption in the line is also taken into account.The solution is presented as the exponential function of the operatorA and the functional calculus is developed. The eigenfunction and the expansion coefficients, in terms of which the explicit solution is expressed, have been found. The nonlinear equation and the explicit expressions for theX-function are derived. The albedo problem with the determined expansion coefficients and the intensity of the emergent radiation is given as an example.     

Micro-structured magnetic atmospheres

Asymmetrical Stokes profiles are produced if the photospheric magnetic and velocity fields are structured on scales smaller than the mean-free-path of the photons. Here we put forward a compact analytical expression for the radiative transfer equation in this case. Explicitly, micro-variations of the magnetic field strength and the velocity are considered. The existence of micro-structures might have serious implications on the techniques currently used to measure solar magnetic fields. For example, we show the failure of the relationship employed to calibrate magnetographs.On leave from the Dipartimento di Astronomia e Scienza dello Spazio, Università di Firenze, Largo E. Fermi 5, I-50125 Firenze, Italy     

Polarization of continuum radiation in magnetic atmospheres

A numerical solution is presented for the problem of continuum radiative transfer in a magnetoactive medium. The continuum opacities are calculated in the presence of a strong magnetic field (H=10⁷ G) typical of magnetic white dwarfs. The L.T.E. pure absorption model is assumed for calculating the polarized radiation field emitted by a realistic model atmosphere in the plane parallel approximation. The wavelength dependence of the linear and circular polarizations are calculated for both uniform and dipole field configurations.     

Net circular polarization in magnetic spectral lines produced by velocity gradients: Some analytical results

The net circular polarization in a spectral line due to the combined effect of magnetic fields and velocity gradients is analyzed for a few schematic situations. In some particular cases, its dependence on the magnetic field, velocity field and line parameters can be expressed analytically.On leave from Dipartimento di Astronomia e Scienza dello Spazio, Università di Firenze, Largo E. Fermi 5, I-50125 Firenze, Italia     

On nonadiabatic waves in the photospheres of cool stars

The coupled set of equations of hydrodynamics and radiative transfer is derived for small disturbances in a plane, grey atmosphere. Only radiative transfer is taken into account in the energy equation; dynamical effects of radiation are ignored. A mean stationary radiative flux through the photosphere is taken into account. The radiative transfer equation is used by assuming the Eddington approximation, moreover, an exponential height profile of the temperature and an analytical opacity formula are supposed. For this model we obtained an asymptotic solution for plane nonadiabatic acoustic waves and radiation waves. The approach provides a detailed discussion of the interaction of nonadiabatic p‐modes and radiation waves in a realistic model of the photosphere of a solar‐like star.     

A generalized theory for line formation in a homogeneous magnetic field

The formation of spectral lines in a homogeneous magnetic field has been studied. A new method for solving the transfer equations for polarized light has been found. Using this method, the Stokes parameters may be derived without any special assumptions regarding the model atmosphere. With the line formed by pure absorption, the expressions for the Stokes parameters may easily be adapted to numerical calculations. In order to illustrate the method, the line profile for the Zeeman triplet 5250 has been calculated using a photospheric model atmosphere.     

Effect of continuum absorption on the polarization of resonance lines

Multiple resonance scattering of radiation in a spectrum line with absorption in the continuum is examined. It is assumed that the scattering atmosphere is semi-infinite and that there is no magnetic field or continuum emission at the frequencies of the spectrum line. The polarization characteristics of the emerging radiation are determined for unpolarized primary radiation sources distributed uniformly within an atmosphere in the case of a Voigt absorption profile. The calculations employ an iterative solution of the Ambartsumyan-Chandrasekhar matrix integral equation.     

Hard X-ray bremsstrahlung produced by electrons escaping a high-temperature thermal source in a solar flare

The problem of production of flare hard X-rays by bremsstrahlung from hot thermal escaping electrons (Skrynnikov and Somov, 1982) in a chromospheric plasma is studied.The Landau kinetic equation is solved near the thermal source of energized electrons in a homogeneous magnetic tube to compute the anisotropic inhomogeneous distribution of the thermal escaping electrons.The intensity and polarization of hard X-rays is also computed and a comparison of theoretical results with observational data is made.On leave from: Istituto di Astronomia, Largo E, Fermi 5, I-50125 Firenze, Italy.     

Numerical solution of MHD free-convection flow in the Stokes problem by the Crank-Nicolson method

A numerical solution of magnetohydrodynamic free-convection flow, in the Stokes' problem, for a porous vertical plate, is obtained, when Prandtl numberP is not equal to one. The magnetic lines of force are assumed to be fixed relative to the plate which is started moving impulsively in its own plane (I.S.P.) or it is uniformly accelerated (U.A.P.). The solution is obtained by the Crank-Nicolson method, which is an implicit second-order method, forP=0.71 (air) andP=7 (water). The obtained results are shown on figures and tables.     

Polarized line formation by resonance scattering: Lorentz profile

Multiple resonance scattering of spectral line radiation is examined in atmospheres with uniformly distributed sources of unpolarized radiation. It is assumed that the profile of the absorption coefficient is lorentzian and that scattering involves complete frequency redistribution. The polarization characteristics of the emerging radiation are determined by iterative solution of a nonlinear Ambartsumyan-Chandrasekhar matrix integral equation. In particular, it is found that for pure scattering the maximum polarization at the limb of the disk is 5.421%. The polarization characteristics of the emerging radiation are compared for three different absorption profiles: Lorentz, Doppler, and rectangular (monochromatic radiation). __________ Translated from Astrofizika, Vol. 50, No. 2, pp. 199–217 (May 2007).     

Hα emission line profiles of solar flares with turbulent velocity fields

The H line profile in a flare atmospheric model superposed by a spatially correlated velocity field is studied in detail in this paper. The computations are carried out with the assumption that the velocity field is represented by a Kubo-Anderson process. It is found that the shape and the intensity of the H line profile depend greatly on the parameters of the velocity field. The variation of the profile and its relative deviation with respect to different correlation lengths is more complex than in the case of absorption lines. It is also shown that such a profile cannot be matched by those produced in the usually-used micro- and macro-turbulent approaches, especially for the velocity field with an intermediate correlation length. The results imply that the flare atmosphere should be highly turbulent in order to explain the observed H line profile with only weak central reversal in the flare spectra. Particularly, the effects of meso-turbulent fields should be taken into account in order to improve the spectral diagnostics and modelling of the flare atmosphere.     

Volterra equation for the matrix source function at resonance scattering

A matrix transfer equation for multiple resonance scattering of radiation in a spectral line in a semiinfinite atmosphere with a uniform distribution of primary radiation sources is examined. A nonlinear matrix integral is obtained for this equation as a generalization of the Rybicki two-point Q-integral. One special case of the matrix [^(Q)] {\mathbf{\hat{Q}}} -integral is the Volterra equation for the matrix source function of the problem discussed here. The Volterra equation is solved numerically for a Doppler profile of the absorption coefficient. Several polarization characteristics of the emerging radiation are obtained.     

Towards a better dynamic theory for type III radio bursts

An approximate numerical solution is given to the quasilinear equations governing the interaction of the electron streams exciting type III solar radio bursts and the background plasma. The density and temperature variations are taken into account and the stream is studied from r =1 to 20 R . The method used is to level any positive slopes in the distribution function as they develop and record the energy released. None of the energy released is allowed to return to the stream and since no other damping mechanisms are taken into account, the solution is only applicable to the initial rise of the burst. Even then, the solution gives more decleration of the leading edge of the burst than observed which indicates that letting some of the energy released return to the stream is necessary. At the same time the solution does show that depletion of high velocity electrons is an important process and calls into question any analysis which neglects this process by, for example, integrating over the spectrum of plasma waves such as Zaitsev et al. (1972). It is shown that the case considered here is an opposite limiting case to the one considered by Zaitsev et al. and what needs to be done to find the more realistic solution in between these two limiting cases is indicated.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

Structure of magnetogasdynamic weak shock waves

The problem of the detailed structure of magnetogasdynamic shock waves is investigated. It is assumed that the flow takes place under normal magnetic fieldH ₀ and the conductivity of the medium is considered infinite. An approximate analytical solution of the nonlinear differential equations describing the phenomena is obtained. The suggested analytical results in this paper are in good agreement with the previous numerical computations for the thickness and the velocity distribution inside the transition region. In addition, the enthalpy distribution inside the shock front is predicted.     

Radio maps of the Sun at λ = 1.95 cm

Two radio maps of the Sun at = 1.95 cm, taken at the N.R.A.O. (Green Bank) with a resolving power of 2.1 are studied and compared with H pictures: both bright and dark features appear to be strongly correlated. The emitted flux and the angular extent of the bright regions are given; from the observed radio-darkening of a region, the height of a prominence is derived.On leave from Osservatorio Astrofisico di Arcetri, Firenze.