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It is shown from the equation of radiative transfer that the visual attenuation of an optically thick, spherically symmetric circumstellar cloud of dust can be small compared to that produced by an equal optical depth of interstellar dust. The reasons for this are the moderately large albedos of the grains and their tendency to emphasize forward scattering.
The electromagnetic and gravitational Stokes parameters are defined in the general theory of relativity. The general-relativistic equation of radiative transfer for polarized radiation is then derived in terms of the Stokes parameters for both high-frequency electromagnetic and gravitational waves. The concept of Stokes parameters is generalized for the most general class of metric theories of gravity, where six (instead of two) independent states of polarization are present.
Results from numerical computations of the diffuse sky radiances and degree of polarization are given. These results were obtained by solving the equation of radiative transfer in the formulation of Eschelbach. In these computations, multiple scattering and absorption by aerosol particles were considered. In addition, results from experimental measurements of the complex refractive index of aerosol samples at different relative humidities are presented. From the radiation flux divergences which were computed based on the determined properties of atmospheric aerosol particles, atmospheric heating rates were derived which were found to be comparable to the heating rates by water vapor.
The intensity of sunlight multiply scattered in model atmospheres is derived from the equation of radiative transfer by an analytical small-angle approximation. The approximate analytical solutions are compared to rigorous numerical solutions of the same problem. Results obtained from an aerosol-laden model atmosphere are presented. Agreement between the rigorous and the approximate solutions is found to be within a few per cent. The analytical solution to the problem which considers an aerosol-laden atmosphere is then inverted to yield a phase function which describes a single scattering event at small angles. The effect of noisy data on the derived phase function is discussed.
Line profiles of 36 multiplets of silicon (Si) II, III, and IV were computed for a grid of model atmospheres covering the range from 15,000 to 35,000 K in effective temperature and 2.5 to 4.5 in log (gravity). The computations involved simultaneous solution of the steady-state statistical equilibrium equations for the populations and of the equation of radiative transfer in the lines. The variables were linearized, and successive corrections were computed until a minimal accuracy of 1/1000 in the line intensities was reached. The common assumption of local thermodynamic equilibrium (LTE) was dropped. The model atmospheres used also were computed by non-LTE methods. Some effects that were incorporated into the calculations were the depression of the continuum by free electrons, hydrogen and ionized helium line blocking, and auto-ionization and dielectronic recombination, which later were found to be insignificant. Use of radiation damping and detailed electron (quadratic Stark) damping constants had small but significant effects on the strong resonance lines of Si III and IV. For weak and intermediate-strength lines, large differences with respect to LTE computations, the results of which are also presented, were found in line shapes and strengths. For the strong lines the differences are generally small, except for the models at the hot, low-gravity extreme of our range. These computations should be useful in the interpretation of the spectra of stars in the spectral range B0-B5, luminosity classes III, IV, and V.
The adding method to solve the equation of radiative transfer has been applied to a sunlit haze above a reflecting surface. The method was modified to include the effect of inhomogeneities on the underlying surface whose dimensions are less than the mean free path of the photons in the haze. Numerical examples of surface-feature contrast degradation due to haze are presented. This analysis renders it feasible to determine the optical thickness of hazes from haze-degraded images of the ground and also provides a technique to restore ground-image contrast, which has been obscured by haze, in photoelectrically scanned images.
The ionization structure of hydrogen in a cloud-containing low-density gas surrounding an O star is calculated, emphasizing how UV photons cause the ionization of hydrogen in regions which are totally obscured from the star by neutral clouds. Particular consideration is given to the ionization of such obscured regions by diffuse Lyman continuum radiation produced when H(+) recombines directly to the ground state. The assumed physical structure of the cloud-containing neighborhood of an O star is discussed, a method is outlined for expressing the emission and absorption properties of the neutral clouds and their shadows in terms of effective volume emissivities and absorption coefficients on a line of sight, and equations of radiative transfer are derived for diffuse and stellar radiation in a cloud-containing H II region. Reradiation of Lyman continuum photons from the ionized boundary layer of a neutral cloud or shadow is analyzed. The expected emission measure across a large cloud-containing H II region is determined and found to compare well with that obtained from observations of H-alpha emission around runaway O stars. It is shown that the hydrogen in cloud shadows may be completely ionized by diffuse radiation within approximately half the Stroemgren radius, depending on the extent of that radius, the average frequency of the Lyman continuum radiation from the star, and the average cloud size.
Surface-based observations of downwelling microwave thermal emission are related to temperature and humidity profiles via a standard integral equation of radiative transfer. Both in clear and in cloudy atmospheres, statistical inversion techniques are used to retrieve profiles from a data vector of brightness observations and surface meteorological constraints. For the clear case, accuracy predictions and profile retrievals are illustrated for: (1) single frequency angular scanned data; (2) multi-frequency angular scanned data; and (3) multi-frequency zenith data. For the last case predicted and achieved accuracies were compared in a recently conducted radiometric experiment. Retrievals of cloud contaminated radiometric data are elaborated.
Solutions are presented for the stagnation-region shock-layer equations, including radiative transfer with spectral lines and silica ablation during Jovian entry. Results for variations of entry angle, sphere-cone configuration, and atmospheric model are given. The effect of silica ablation on the radiative and convective surface heating is correlated with the ratio of the wall to free-stream mass flux. Correlations are also given for spectral distributions. The effect of newly obtained SiO radiation properties on the surface heating is examined.
An experiment has been proposed to measure the zenith sky intensity and directly transmitted solar flux density at several wavelengths in the near ultraviolet from an aircraft platform located at about 15 km. From these measurements, the vertical distribution of ozone above the platform is to be inferred, by inversion of the equation of radiative transfer. The present study is an analysis of the errors to be expected in a real experiment, due to atmospheric fluctuations, numerical procedures, and ignorance of various parameters necessary for the calculation of the intensities and fluxes to be expected. These errors are found to vary from 17% at the shorter wavelengths to about 3% at the longer wavelengths used. It is shown elsewhere (Twomey et al., 1977) that unacceptable levels of instability and non-uniqueness of results will be found if the error levels in the measurements upon which we wish to invert exceed 3%. On the basis of this, it is concluded that with the present instrumentation and the present knowledge of necessary ancillary parameters, this experiment will not be feasible.
The paper presents a model for contrast reduction by atmospheric haze developed for the 'two-halves' field of the earth's surface and other geometries of the earth's surface albedo. The model is based on a simplified solution of the equation of radiative transfer in two dimensions, resulting in a method for calculation of the upward zenith intensity in the atmosphere as a function of the distance from the border between the two half planes, for an unabsorbing atmosphere. The adjacency effect between two infinitesimal areas of different albedos is calculated; the resultant simplified solution is used to develop expressions for the line-spread function of the atmosphere and the modulation transfer function. The line-spread function is used to calculate the point spread function, which can be used to compute the intensity above any surface with given spatial dependence of the reflectivity.
The cyclotron absorption coefficients in the ordinary and extraordinary modes are calculated for the shock heated region of AM Her. The equations of radiative transfer are solved and the intensity of the emitted UV radiation determined as a function of angle. The average spectrum is shown to have deviations from the previously predicted Rayleigh-Jeans spectrum and the magnetic field of AM Her is deduced to be roughly 5 x 10 to the 7th power gauss.
Modified radiative transfer (MRT) equations appropriate for electromagnetic wave propagation in bounded random media are derived from the Bethe-Salpeter equation with the ladder approximation and the Dyson equation with the nonlinear approximation. The MRT equations are then solved with the first-order renormalization approximation to obtain analytical results for the backscattering cross sections of a two-layer random medium with arbitrary three-dimensional correlation functions. The coherent effects of the MRT theory are illustrated and comparisons are made with backscattering cross sections obtained with the first Born approximation to the wave equation.
The effects of an albedo of internal dust, such as ionization structure and temperature of dust grain, were studied by the quasi-diffusion method with an iterative technique for solving the radiative heat transfer equations. It was found that the generalized on-the-spot approximation solution is adequate for most astrophysical applications for a zero albedo; for a nonzero albedo, the Eddington approximation is more accurate. The albedo increases the average energy of the diffuse photons, increasing the ionization level of hydrogen and heavy elements if the Eddington approximation is applied; the dust thermal gradient is reduced so that the infrared spectrum approaches blackbody spectrum with an increasing albedo.
A procedure is developed for calculating atmospheric extinction characteristics (optical thickness, scattering height, single scattering albedo) and surface albedo from radiometric images made at satellite altitudes. The procedure - a fast, computerized method - is suitable for the high-volume processing of satellite imagery data and thus can be used to map temporal and spatial distributions of aerosol parameters. Based on an analytical approximate solution to the equation of radiative transfer in a plane parallel atmosphere, the procedure is primarily applicable to the 0.4-micron to 0.8-micron wavelength range for solar zenith angles from 10 deg to 60 deg, surface albedos between 0.03 and about 0.5, and atmospheric optical thicknesses from 0.2 to 2. With a step-like change in the surface albedo (for example, a seashore or river bank or other similar change of terrain), both surface reflectivity and atmospheric optical thickness can be derived from radiance measurements. In this case, the resultant optical thickness is not based on a known surface reflectivity and is in essence independent of the radiometer calibration.
Methods of regularized and generalized cross validation can be used to estimate the atmosphere's temperature, moisture, and wind structure from a finite number m noisy measurements by meteorological satellites on the intensity of upwelling radiation in selected channel frequencies. The inversion of the equation of radiative transfer is discussed for data obtained by TIROS N satellite.
The process of ablation is calculated for the stagnation region of a flat disk in a radiation-dominated, massive-blowing environment produced in a ballistic range filled with argon. Flow environments are determined by solving the boundary-layer equations while radiative transfer is calculated through a line-by-line spectral computation. The resulting wall heat-transfer rates are coupled with an existing material's response code to determine surface recession and char thickness. The calculation is performed for six 5-cm-diam models made of carbon-phenolic and carbon-carbon composite launched in the Track-G facility at the Arnold Engineering Development Center. Significant surface recessions are predicted to occur for these models due mostly to radiative heating.
A numerical algorithm based on Eddington's second approximation to the equation of radiative transfer has been developed in order to compute the radiances with horizontal and vertical polarization that emerge from precipitating clouds. This algorithm yields a rapid solution to problems pertaining to clouds with vertically inhomogeneous structure. Precipitating clouds containing liquid, mixtures of phases, and ice hydrometeors are modelled. It is shown that the lower-frequency radiances are sensitive to liquid precipitation at low altitudes while the higher-frequency radiances are more sensitive to the ice hydrometeors at the cloud tops. The extinction coefficients of aspherical hydrometeors are presented as a function of rainfall rates. The fact that vertically polarized radiances are warmer than horizontally polarized ones at high rainfall rates, with the difference diminishing at lower frequencies, is due to aspherical ice hydrometeors in the upper regions of precipitating clouds.