Plane-parallel Atmosphere Research Articles

Multi sky-view 3D aerosol distribution recovery

Aerosols affect climate, health and aviation. Currently, their retrieval assumes a plane-parallel atmosphere and solely vertical radiative transfer. We propose a principle to estimate the aerosol distribution as it really is: a three dimensional (3D) volume. The principle is a type of tomography. The process involves wide angle integral imaging of the sky on a very large scale. The imaging can use an array of cameras in visible light. We formulate an image formation model based on 3D radiative transfer. Model inversion is done using optimization methods, exploiting a closed-form gradient which we derive for the model-fit cost function. The tomography model is distinct, as the radiation source is unidirectional and uncontrolled, while off-axis scattering dominates the images.

Spherically symmetric model stellar atmospheres and limb darkening

Limb darkening is a fundamental ingredient for interpreting observations of planetary transits, eclipsing binaries, optical/infrared interferometry and microlensing events. However, this modeling traditionally represents limb darkening by a simple law having one or two coefficients that have been derived from plane-parallel model stellar atmospheres, which has been done by many researchers. More recently, researchers have gone beyond plane-parallel models and considered other geometries. We previously studied the limb-darkening coefficients from spherically symmetric and plane-parallel model stellar atmospheres for cool giant and supergiant stars, and in this investigation we apply the same techniques to FGK dwarf stars. We present limb-darkening coefficients, gravity-darkening coefficients and interferometric angular diameter corrections from Atlas and SAtlas model stellar atmospheres. We find that sphericity is important even for dwarf model atmospheres, leading to significant differences in the predicted coefficients.

A New High-Precision Correction Method of Temperature Distribution in Model Stellar Atmospheres

Abstract The main features of the temperature correction methods, suggested and used in modeling of plane-parallel stellar atmospheres, are discussed. The main features of the new method are described. Derivation of the formulae for a version of the Unsöld-Lucy method, used by us in the SMART (Stellar Model Atmospheres and Radiative Transport) software for modeling stellar atmospheres, is presented. The method is based on a correction of the model temperature distribution based on minimizing differences of flux from its accepted constant value and on the requirement of the lack of its gradient, meaning that local source and sink terms of radiation must be equal. The final relative flux constancy obtainable by the method with the SMART code turned out to have the precision of the order of 0.5 %. Some of the rapidly converging iteration steps can be useful before starting the high-precision model correction. The corrections of both the flux value and of its gradient, like in Unsöld-Lucy method, are unavoidably needed to obtain high-precision flux constancy. A new temperature correction method to obtain high-precision flux constancy for plane-parallel LTE model stellar atmospheres is proposed and studied. The non-linear optimization is carried out by the least squares, in which the Levenberg-Marquardt correction method and thereafter additional correction by the Broyden iteration loop were applied. Small finite differences of temperature (δT/T = 10−3) are used in the computations. A single Jacobian step appears to be mostly sufficient to get flux constancy of the order 10−2 %. The dual numbers and their generalization – the dual complex numbers (the duplex numbers) – enable automatically to get the derivatives in the nilpotent part of the dual numbers. A version of the SMART software is in the stage of refactorization to dual and duplex numbers, what enables to get rid of the finite differences, as an additional source of lowering precision of the computed results.

New Methods in Modeling of Hot Stellar Atmospheres

Abstract In the present study we had three main aims. First to study the possibility of reducing the initial model atmosphere data to short analytical polynomials. The second was to use as the depth variable the logarithm of the local gas pressure instead the Rosseland mean. The third aim was to check the applicability of the derived formulae and proposed computation methods to obtain high precision self-consistent results in modeling hot plane-parallel stellar atmospheres. Introducing the dimensionless (reduced) local quantities θ = T/T eff and β = P/P(T eff) it has been shown that for hot convection-free stellar atmospheres the curves log θ versus log β reduce an initial grid of models to simple polynomials and bring forth some general features of the model stellar atmospheres. Even for stellar atmospheres having the convective zones in the deeper atmospheric layers, the outer part of the atmosphere (up to T = T eff and for T eff > 5000 K) can be described in the same manner by curves log θ versus log β as for the hotter stars. Iterative modeling of any hot stellar atmosphere can be started from these formulae (obtained for solar abundances), using rational polynomial ratios for P(T eff), obtaining from these data the needed T versus P dependence. To check suitability of the formulae, the iterative correction of the model stellar atmospheres has been carried out by the traditional Unsöld-Lucy method and by the novel least squares optimization based on Levenberg-Marquardt method, followed by Broyden correction loop. It has been shown that the flux constancy obtained by it is almost 2 dex higher than obtained by the Unsöld-Lucy method. The precision estimators as criteria of the modeling algorithms self-consistency and of the computational precision level have been proposed and used.

Theoretical formulation of Doppler redistribution in scattering polarization within the framework of the velocity-space density matrix formalism

Within the framework of the density matrix theory for the generation and transfer of polarized radiation, velocity density matrix correlations represent an important physical aspect that, however, is often neglected in practical applications by adopting the simplifying approximation of complete redistribution on velocity. In this paper, we present an application of the Non-LTE problem for polarized radiation taking such correlations into account through the velocity-space density matrix formalism. We consider a two-level atom with infinitely sharp upper and lower levels, and we derive the corresponding statistical equilibrium equations neglecting the contribution of velocity-changing collisions. Coupling such equations with the radiative transfer equations for polarized radiation, we derive a set of coupled equations for the velocity-dependent source function. This set of equations is then particularized to the case of a plane-parallel atmosphere. The equations presented in this paper provide a complete and solid description of the physics of pure Doppler redistribution, a phenomenon generally described within the framework of the redistribution matrix formalism. The redistribution matrix corresponding to this problem (generally referred to as R_I) is derived starting from the statistical equilibrium equations for the velocity-space density matrix and from the radiative transfer equations for polarized radiation, thus showing the equivalence of the two approaches.

Spectral modelling of massive binary systems

Aims: We simulate the spectra of massive binaries at different phases of the orbital cycle, accounting for the gravitational influence of the companion star on the shape and physical properties of the stellar surface. Methods: We used the Roche potential modified to account for radiation pressure to compute the stellar surface of close circular systems and we used the TIDES code for surface computation of eccentric systems. In both cases, we accounted for gravity darkening and mutual heating generated by irradiation to compute the surface temperature. We then interpolated NLTE plane-parallel atmosphere model spectra in a grid to obtain the local spectrum at each surface point. We finally summed all contributions, accounting for the Doppler shift, limb-darkening, and visibility to obtain the total synthetic spectrum. We computed different orbital phases and sets of physical and orbital parameters. Results: Our models predict line strength variations through the orbital cycle, but fail to completely reproduce the Struve-Sahade effect. Including radiation pressure allows us to reproduce a surface temperature distribution that is consistent with observations of semi-detached binary systems. Conclusions: Radiation pressure effects on the stellar surface are weak in (over)contact binaries and well-detached systems but can become very significant in semi-detached systems. The classical von Zeipel theorem is sufficient for the spectral computation. Broad-band light curves derived from the spectral computation are different from those computed with a model in which the stellar surfaces are equipotentials of the Roche potential scaled by the instantaneous orbital separation. In many cases, the fit of two Gaussian/Lorentzian profiles fails to properly measure the equivalent width of the lines and leads to apparent variations that could explain some of the effects reported in the literature.

Solution of Linear Radiative Transfer Problems in Plane-Parallel Atmospheres. III

Solution of Linear Radiative Transfer Problems in Plane-Parallel Atmospheres. III

Influence of departures from LTE on oxygen abundance determination in the atmospheres of A-K stars

We have performed non-LTE calculations for O I with a multilevel model atom using currently available atomic data for a set of parameters corresponding to stars of spectral types from A to K. Departures from LTE lead to a strengthening of O I lines, and the difference between the non-LTE and LTE abundances (non-LTE correction) is negative. The non-LTE correction does not exceed 0.05 dex in absolute value for visible O I lines for main-sequence stars in the entire temperature range. For the infrared O I 7771 A line, the non-LTE correction can reach −1.9 dex. The departures from LTE are enhanced with increasing temperature and decreasing surface gravity. We have derived the oxygen abundance for three A-type mainsequence stars with reliably determined parameters (Vega, Sirius, HD 32115). For each of the stars, allowance for the departures from LTE leads to a decrease in the difference between the abundances from infrared and visible lines, for example, for Vega from 1.17 dex in LTE to 0.14 dex when abandoning LTE. In the case of Procyon and the Sun, inelastic collisions with HI affect the statistical equilibrium of OI, and agreement between the abundances from different lines is achieved when using Drawin’s classical formalism. Based on the O I 6300, 6158, 7771-5, and 8446 A lines of the solar spectrum, we have derived the mean oxygen abundance log ɛ = 8.74 ± 0.05 using a classical plane-parallel model solar atmosphere and log ɛ +3D = 8.78 ± 0.03 by applying the 3D corrections taken from the literature.

THE STABILITY OF WEAKLY COLLISIONAL PLASMAS WITH THERMAL AND COMPOSITION GRADIENTS

Over the last decade, substantial efforts have been devoted to understanding the stability properties, transport phenomena, and long-term evolution of weakly-collisional, magnetized plasmas which are stratified in temperature. These studies have improved our understanding of the physics governing the intra-cluster medium (ICM), but assumed that ICM is a homogeneous. This, however, might not be a good approximation if heavy elements sediment in the inner region of the galaxy cluster. In this paper, we analyze the stability of a weakly-collisional, magnetized plane-parallel atmosphere which is stratified in both temperature and composition. This allows us to discuss for the first time the dynamics of weakly-collisional environments where heat conduction, momentum transport, and ion-diffusion are anisotropic with respect to the direction of the magnetic field. We show that, depending on the relative signs and magnitudes of the gradients in the temperature and the mean molecular weight, the plasma can be subject to a wide variety of unstable modes which include modifications to the magnetothermal instability (MTI), the heat-flux-driven buoyancy instability (HBI), and overstable gravity modes previously studied in homogeneous media. We discuss the astrophysical implications of our findings for a representative galaxy cluster where helium has sedimented. Our findings suggest that the core insulation that results from the magnetic field configurations that arise as a natural consequence of the HBI, which would be MTI stable in a homogeneous medium, could be alleviated if the mean molecular weight gradient is steep enough, i.e., $(\nabla \mu)/\mu > (\nabla T)/T$. This study constitutes a first step toward understanding the interaction between magnetic turbulence and the diffusion of heavy elements in the ICM. (abridged)

Retrieval of aerosol optical depth over land based on a time series technique using MSG/SEVIRI data

Abstract. A novel approach for the joint retrieval of aerosol optical depth (AOD) and aerosol type, using Meteosat Second Generation – Spinning Enhanced Visible and Infrared Imagers (MSG/SEVIRI) observations in two solar channels, is presented. The retrieval is based on a Time Series (TS) technique, which makes use of the two visible bands at 0.6 μm and 0.8 μm in three orderly scan times (15 min interval between two scans) to retrieve the AOD over land. Using the radiative transfer equation for plane-parallel atmosphere, two coupled differential equations for the upward and downward fluxes are derived. The boundary conditions for the upward and downward fluxes at the top and at the bottom of the atmosphere are used in these equations to provide an analytic solution for the AOD. To derive these fluxes, the aerosol single scattering albedo (SSA) and asymmetry factor are required to provide a solution. These are provided from a set of six pre-defined aerosol types with the SSA and asymmetry factor. We assume one aerosol type for a grid of 1°×1° and the surface reflectance changes little between two subsequent observations. A k-ratio approach is used in the inversion to find the best solution of atmospheric properties and surface reflectance. The k-ratio approach assumes that the surface reflectance is little influenced by aerosol scattering at 1.6 μm and therefore the ratio of surface reflectances in the solar band for two subsequent observations can be well-approximated by the ratio of the reflectances at 1.6 μm. A further assumption is that the surface reflectance varies only slightly over a period of 30 min. The algorithm makes use of numerical minimisation routines to obtain the optimal solution of atmospheric properties and surface reflectance by selection of the most suitable aerosol type from pre-defined sets. A detailed analysis of the retrieval results shows that it is suitable for AOD retrieval over land from SEVIRI data. Six AErosol RObotic NETwork (AERONET) sites with different surface types are used for detailed analysis and 42 other AERONET sites are used for validation. From 445 collocations representing stable and homogeneous aerosol type, we find that >75% of the MSG-retrieved AOD at 0.6 and 0.8 μm values compare favourably with AERONET observed AOD values, within an error envelope of ± 0.05 ± 0.15 τ and a high correlation coefficient (R>0.86). The AOD datasets derived using the TS method with SEVIRI data is also compared with collocated AOD products derived from NASA TERRA and AQUA MODIS (The Moderate-resolution Imaging Spectroradiometer) data using the Dark Dense Vegetation (DDV) method and the Deep Blue algorithms. Using the TS method, the AOD could be retrieved for more pixels than with the NASA Deep Blue algorithm. This method is potentially also useful for surface reflectance retrieval using SEVIRI observations. The current paper focuses on AOD retrieval and analysis, and the analysis and validation of reflectance will be given in a following paper.

Analysis of optical spectra of V1357 Cyg≡Cyg X-1

Optical spectra and light curves of the massive X-ray binary V1357 Cyg are analyzed. The calculations were based on models of irradiated plane-parallel stellar atmospheres, taking into account reflection of the X-ray radiation, asphericity of the stellar surface, and deviations from LTE for several ions. Comparison of observed spectra obtained in 2004–2005 at the Bohyunsan Observatory (South Korea) revealed variations of the depths of HI lines by up to 18% and of HeI and heavy elements lines by up to 10%. These variations are not related to the orbital motion of the star, and are probably due to variations of the stellar wind intensity. Perturbations of the thermal structure of the atmosphere due to irradiation in various states of Cyg X-1 (including outburst) do not lead to the formation of a hot photosphere with an electron temperature exceeding the effective temperature. As a result, variations of the profiles of optical lines of HI, HeI, and heavy elements due to the orbital motion of the star and variations of the irradiating X-ray flux do not exceed 1% of the residual intensities. Allowing for deviations from LTE enhances the HI and HeI lines by factors of two to three and the MgII lines by a factor of nine, and is therefore required for a fully adequate analysis of the observational data. Analysis of the HI, HeI, and HeII lines profiles yielded the following set of parameters for theOstar at the observing epoch: T eff = 30 500±500 K, log g = 3.31±0.05, [He/H] = 0.42 ± 0.05. The observed HeI line profiles have emission components that are formed in the stellar wind and increase with the line intensity. The abundances of 11 elements in the atmospheres of V1357 Cyg and α Cam, which has a similar spectral type and luminosity class, are derived. The chemical composition of V1357 Cyg is characterized by a strong excess of helium, nitrogen, neon, and silicon, which is related to the binarity of the system.

The nature of the high Galactic latitude O-star HD 93521: new results from X-ray and optical spectroscopy

Owing to its unusual location and its isolation, the nature of the high Galactic latitude O9.5Vp object HD93521 is still uncertain. We have collected X-ray and optical observations to characterize the star and its surroundings. X-ray images and spectra are analyzed to search for traces of a recent star formation event around HD93521 and to search for the signature of a possible compact companion. Optical echelle spectra are analysed with plane-parallel model atmosphere codes, assuming either a spherical star or a gravity darkened rotationally flattened star, to infer the effective temperature and surface gravity, and to derive the He, C, N and O abundances of HD93521. The X-ray images reveal no traces of a population of young low-mass stars coeval with HD93521. The X-ray spectrum of HD93521 is consistent with a normal late O-type star although with subsolar metallicity. No trace of a compact companion is found in the X-ray data. In the optical spectrum, He and N are found to be overabundant, in line with the effect of rotational mixing in this very fast rotator, whilst C and O are subsolar. A critical comparison with the properties of subdwarf OB stars, indicates that, despite some apparent similarities, HD93521 does not belong to this category. Despite some ambiguities on the runaway status of the star, the most likely explanation is that HD93521 is a Population I massive O-type star that was ejected from the Galactic plane either through dynamical interactions or a result of a supernova event in a binary system.

X-ray bursting neutron star atmosphere models using an exact relativistic kinetic equation for Compton scattering

Theoretical spectra of X-ray bursting neutron star (NS) model atmospheres are widely used to determine the basic NS parameters such as their masses and radii. We construct accurate NS atmosphere models using for the first time an exact treatment of Compton scattering via the integral relativistic kinetic equation. We also compare the results with the previous calculations based on the Kompaneets operator. We solve the radiation transfer equation together with the hydrostatic equilibrium equation accounting exactly for the radiation pressure by electron scattering. We thus construct a new set of plane-parallel atmosphere models in LTE for hot NSs. The models were computed for six chemical compositions (pure H, pure He, solar H/He mix with various heavy elements abundances Z = 1, 0.3, 0.1, and 0.01 Z_sun, and three log g = 14.0, 14.3, and 14.6. For each chemical composition and log g, we compute more than 26 model atmospheres with various luminosities relative to the Eddington luminosity L_Edd computed for the Thomson cross-section. The maximum relative luminosities L/L_Edd reach values of up to 1.1 for high gravity models. The emergent spectra of all models are redshifted and fitted by diluted blackbody spectra in the 3--20 keV energy range appropriate for the RXTE/PCA. We also compute the color correction factors f_c. The radiative acceleration g_rad in our luminous, hot-atmosphere models is significantly smaller than in corresponding models based on the Kompaneets operator, because of the Klein-Nishina reduction of the electron scattering cross-section, and therefore formally "super-Eddington" model atmospheres do exist. The differences between the new and old model atmospheres are small for L / L_Edd < 0.8. For the same g_rad / g, the new f_c are slightly larger (by approximately 1%) than the old values.

Small-angle modification of the radiative transfer equation for a pseudo-spherical atmosphere

The conventional pseudo-spherical technique relies on the separation of the total radiance into the direct solar beam and the diffuse radiance; the direct solar radiance is treated in a spherical geometry, while the diffuse radiance is computed in a plane-parallel geometry. In the small-angle modification of the radiative transfer equation, the total radiance is separated into an anisotropic part and a regular part. In this paper, we pOresent two formulations of the small-angle modification of the radiative transfer equation for a pseudo-spherical atmosphere. In the first formulation, we solve the radiative transfer equation for the diffuse radiance in a pseudo-spherical atmosphere with an additional anisotropic source term computed in a plane-parallel atmosphere, while in the second formulation we solve the radiative transfer equation for the regular solution in a plane-parallel atmosphere with an additional pseudo-spherical correction term. The numerical analysis revealed that the accuracy of the small-angle models is acceptable.

Verification of radiative transfer results by inserting them into the RTE: A demonstration for Rayleigh scattering

The verification of a new or updated radiative transfer model (RTM) is one of the important steps in its development; this is usually achieved by comparisons with real measurements or published tables of generally accepted radiative transfer results. If such tables do not exist, verification becomes more complicated and an external review of the implementation is often unpractical due to the sheer amount and complexity of the code. The presented verification approach is to “simply” insert results of radiative transfer (RT) calculations into the radiative transfer equation (RTE). The evaluation of the RTE consists of numerically calculating partial derivatives and integrals, which is much simpler to implement than a solution of the RTE. Presented is a demonstration of this approach for a case of Rayleigh scattering in a plane parallel atmosphere, which showed only very small deviation from the radiative transfer equation.This approach has two key benefits. First, its implementation into a high level computer language can be very short (≈60 lines in MATHEMATICA) and clear compared to a full RTM; and such code is much more easy to review. Second, this approach can be easily extended to cases where no other independent RT implementation is available for validation. The proposed implementation and data are provided with this paper.

On the Analytical Solution of the SN Radiative Transport Equation in a Slab for a Space-dependent Albedo Coefficient

In this work, we report a genuine general analytical solution for the linearized SN radiative-conductive transfer problem in a heterogeneous plane parallel atmosphere with the albedo coefficient depending continuously on the spatial variable. By general solution, we mean that the solution is valid for an arbitrary albedo coefficient continuous functions of the spatial variable having the property of fulfill the requirements of existence and uniqueness. The key feature of this novel approach embodies the steps: following the idea of the Decomposition method, we transform the original problem into a set of recursive problems with constant albedo coefficients, having the main feature that the sources terms takes the information of the spatial dependency of the albedo coefficient into account. This procedure allows us to solve, analytically, the resulting recursive system by the LTSN method developed for a constant albedo coefficient. Finally, we present the error control analysis of the solution and numerical comparisons against the literature results.

Solution of linear radiative transfer problems in plane-parallel atmospheres. II

Solution of linear radiative transfer problems in plane-parallel atmospheres. II

The HBI in a quasi-global model of the intracluster medium

In this paper, we investigate how convective instabilities influence heat conduction in the intracluster medium (ICM) of cool-core galaxy clusters. The ICM is a high-beta, weakly collisional plasma in which the transport of momentum and heat is aligned with the magnetic field. The anisotropy of heat conduction, in particular, gives rise to instabilities that can access energy stored in a temperature gradient of either sign. We focus on the heat-flux-driven buoyancy instability (HBI), which feeds on the outwardly increasing temperature profile of cluster cool cores. Our aim is to elucidate how the global structure of a cluster impacts on the growth and morphology of the linear HBI modes when in the presence of Braginskii viscosity, and ultimately on the ability of the HBI to thermally insulate cores. We employ an idealized quasi-global model, the plane-parallel atmosphere, which captures the essential physics – e.g. the global radial profile of the cluster – while letting the problem remain analytically tractable. Our main result is that the dominant HBI modes are localized to the innermost (≲20 per cent) regions of cool cores. It is then probable that, in the non-linear regime, appreciable field-line insulation will be similarly localized. Thus, while radio-mode feedback appears necessary in the central few tens of kpc, heat conduction may be capable of offsetting radiative losses throughout most of a cool core over a significant fraction of the Hubble time. Finally, our linear solutions provide a convenient numerical test for the non-linear codes that simulate the saturation of such convective instabilities in the presence of anisotropic transport.

Cepheid limb darkening, angular diameter corrections, and projection factor from static spherical model stellar atmospheres

Context. One challenge for measuring the Hubble constant using Classical Cepheids is the calibration of the Leavitt Law or period-luminosity relationship. The Baade-Wesselink method for distance determination to Cepheids relies on the ratio of the measured radial velocity and pulsation velocity, the so-called projection factor and the ability to measure the stellar angular diameters. Aims. We use spherically-symmetric model stellar atmospheres to explore the dependence of the p-factor and angular diameter corrections as a function of pulsation period. Methods. Intensity profiles are computed from a grid of plane-parallel and spherically-symmetric model stellar atmospheres using the SAtlas code. Projection factors and angular diameter corrections are determined from these intensity profiles and compared to previous results. Results. Our predicted geometric period-projection factor relation including previously published state-of-the-art hydrodynamical predictions is not with recent observational constraints. We suggest a number of potential resolutions to this discrepancy. The model atmosphere geometry also affects predictions for angular diameter corrections used to interpret interferometric observations, suggesting corrections used in the past underestimated Cepheid angular diameters by 3 - 5%. Conclusions. While spherically-symmetric hydrostatic model atmospheres cannot resolve differences between projection factors from theory and observations, they do help constrain underlying physics that must be included, including chromospheres and mass loss. The models also predict more physically-based limb-darkening corrections for interferometric observations.

Symmetry relations revealed in Mueller matrix hemispherical maps

The hemispherical Mueller matrix map for light reflected from a plane-parallel planetary atmosphere is shown to obey several symmetry properties that provide a straightforward method to check their physical realizability. The mirror scattering relation and the reciprocity relation are employed to support the symmetry rules for hemispherical Mueller matrix maps. Additionally, two classes of experiments are identified in which the symmetry rules can be applied, namely, when the incident beam is at zenith and off-zenith angles and the scattering particles are spheres and randomly oriented hexagonal particles.