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1.
Three new approximate forms for theH-function for isotropic scattering have been developed. Each of the forms involve three unknown functions of albedo .  相似文献   

2.
In this work, we first establish a simple procedure to obtain with 11-figure accuracy the values of Chandrasekhar’s H-function for isotropic scattering using a closed-form integral representation and the Gauss-Legendre quadrature. Based on the numerical values of the function produced by this method for various combinations of ? 0, the single scattering albedo, and μ, the cosine of the zenith angle θ of the direction of radiation emergent from or incident upon a semi-infinite scattering-absorbing medium, we propose a rational approximation formula with μ 1/4 and \(\sqrt{1-\varpi_{0}}\) as the independent variables. This allows us to reproduce the correct values of H(? 0,μ) within a relative error of 2.1×10?5 without recourse to any iterative procedure or root-finding process.  相似文献   

3.
We have considered the transport equation for radiative transfer to a problem in semi-infinite non-conservative atmosphere with no incident radiation and scattering albedo 0 < 1. Usint the Laplace transform and the Wiener-Hopf technique, we have determined the emergent intensity and the intensity at any optical depth. We have obtained theH-function of Dasgupta (1977) by equating the emergent intensity with the intensity at zero optical depth.  相似文献   

4.
The first two derivatives and the mixed derivative of theH-function with respect to the angular variable and the albedo of single scattering are studied in the case of isotropic scattering using the kernel approximation method. This allows to obtain simple formulae the accuracy of which is estimated to be very good in rather low orders of approximation. Some samples of numerical results are given with eight decimal accuracy.  相似文献   

5.
The determination of the average path-length of photons emerging from a finite planeparallel atmosphere with molecular scattering is discussed. We examine the effects of polarisation on the average path-length of the emergent radiation by comparing the results with those obtained for the atmosphere where the scattering obeys the scalar Rayleigh function. Only the axial radiation field is considered for both cases.To solve this problem we have used the integro-differential equations of Chandrasekhar for the diffuse scattering and transmission functions (or matrices). By differentiation of these equations with respect to the albedo of single scattering we obtain new equations the solution of which gives us the derivatives of the intensities of the emergent radiation at the boundaries.As in the case of scalar transfer the principles of invariance by Chandrasekhar may be used to find an adding scheme to obtain both the scattering and transmission matrices and their derivatives with respect to the albedo of single scattering. These derivatives are crucial in determining the average path length.The numerical experiments have shown that the impact of the polarisation on the average pathlength of the emergent radiation is the largest in the atmospheres with optical thickness less than, or equal to, three, reaching 6.9% in the reflected radiation.  相似文献   

6.
The apparent strength of absorptions due to H2O near 1.9 and 3 μm in reflectance spectra is strongly affected by sample albedo. This study uses experimental and analytical approaches to quantify the effects of albedo on estimating the water content of hydrated minerals using various band parameters. We compare spectral band parameters for a series of low-albedo physical and numerical mixtures to measured water contents. Physical experiments consist of montmorillonite, clinoptilolite, and palagonite mixed with lesser amounts of carbon black and ilmenite, whereas numerical mixtures are composed of these host minerals mixed with a material of constant, low albedo. We find the effective single-particle absorption-thickness parameter provides the best correlation to water content, independent of composition and albedo, when derived from continuum-removed single scattering albedo spectra. Uncertainties in estimated water content are on the order of ±1 wt% using this method. The normalized optical path length parameter provides the best correlation to water content when using reflectance spectra, yielding estimates within ±1.6 wt% H2O. The accuracy of these models is related to the physical nature of the darkening material. Scattering and absorption efficiencies are easier to model for intimate mixtures containing relatively large, dark grains than mixtures dominated by coatings of a fine-grained, strongly absorbing material. This suggests the physical properties that give rise to the albedo of a material are an important factor for accurate estimates of absolute water content.  相似文献   

7.
The algorithm for determining effective optical thickness of absorption line formation in a plane-parallel homogeneous planetary atmosphere is presented. The case of anisotropic scattering is considered. The results of numerical calculations of τ e (μ0) at the scattering angle γ = π for some values of the single scattering albedo λ and the parameter of the Heyney-Greenstein scattering indicatrix g are given. The refined equation for the function T m (−μ, μ0) is presented.  相似文献   

8.
In this note we derive an exact solution of transfer equation in a plane-parallel semiinfinite atmosphere with albedo >1, by the method of Laplace transform and Wiener-Hopf technique. The emergent intensityI(0, ) is obtained in terms of theH 0-functionH 0() (Das Gupta, 1978) for which some good approximations are given. Intensity at any depth is also obtained.I(0, )/I(0, 0) is plotted in graphs against [0,1], and shows a maximum which drops and shifts towards the origin as increases.  相似文献   

9.
The purpose of this paper is to present a numerical technique to directly compute the Chandrasekhar'sH ()-function for anisotropic scattering in terms of the roots of the characteristic equations as well as the quadrature points of a certain degreen employed to approximate the definite integral involved in the basic equation. The principal feature of the algorithm proposed here is a compact computer code to enumerate n C m combinations ofn distinct integers {1,...,n} takenm at a time. With these quantities available, the coefficients of the polynomial equation of the characteristics equation can be readily computed for any given characteristic function, so that a standard technique such as the Laguerre method can be applied to find all the roots.It is shown that the results obtained for some representativeH()-functions using the present technique with relatively low-order formula (e.g.,n=7) are sufficiently accurate for all practical purposes.  相似文献   

10.
Extinction and scattering efficiencies are calculated for spherically stratified graphite spheres using formulae which are valid in the small particle limit. The resulting extinction curve for this model is shown to peak at an ultraviolet wavelength -1=4.8–5 -1 close to that for the case of the less probable plane stratified model of a graphite sphere. The albedo in the present model is higher than that calculated for the plane stratified case by a factor 1.5–2 in the far ultraviolet. Extinction curves are also obtained for the case of a dielectric sphere surrounded by a thin graphite film, and it is shown that there now is an extinction minimum at -14.6 -1.  相似文献   

11.
By performing the one-sided Laplace transform on the scalar integro-differential equation for a semi-infinite plane-parallel isotropic scattering atmosphere with a scattering albedo 0 1, an integral equation for the emergent intensity has been derived. Application of the Wiener-Hopf technique to this integral equation will give the emergent intensity. The intensity at any optical depth for a positive scattering angle is also derived by inversion. The intensity at any optical depth for a negative scattering angle is also derived in terms of Cauchy's principal value using Plemelj's formulae.  相似文献   

12.
This study considers the influence of the effects of scattering due to Langmuir turbulent pulsations in the transfer of radiation in the spectral lines. The transfer equation of radiation in spectral lines, by taking into account scattering due to Langmuir turbulent pulsations, is written in a form convenient for application by numerical methods.The profile's intensity for a plane-parallel finite isothermal slab of a turbulent plasma in the case of complete redistribution of scattering by an atom are obtained. Numerical studies show that in this case with the broadening of spectral lines and the decreasing of self-reversal, the Langmuir frequency pe is of the same order as the electronic Doppler width De. Creation of the line satellites when pe is larger than the line width is shown with the aid of numerical methods.  相似文献   

13.
To facilitate the computation of the radiative intensity reflected, upon multiple scattering, by a vertically inhomogeneous medium, an implicit formula for integrating the invariant imbedding equation for Fourier-decomposed reflection function is derived starting with the formal solution: the height variations of the single-scattering albedo and the phase function characterizing the degree of inhomogeneity are thereby approximated by piece-wise, but continuous, linear functions of optical height , while the reflection function is approximated by a piece-wise quadratic polynomial in over each integration step. Using these approximations, the integration involved in the formal solution is then carried out analytically, yielding a correctortype formula for finding the reflection function at each step of . It is expected that this formula is capable of handling general cases of inhomogeneous media where both single-scattering albedo and phase function are allowed to vary continuously with height.Similar, but explicit expressions are also derived for the single and the second-order scattering solutions, with which the higher-order Fourier terms of reflection function are to be approximated, thereby enabling us to avoid the iterative process.  相似文献   

14.
In this paper of the series, the expansions of the functionsH 1,H 2, andH 3 will be established analytically and computationally form positive integer,q any real number and , are both positive <1. Full recursive computational algorithms with their numerical results will also be included.  相似文献   

15.
The numerical solution of radiative transfer equation including curvature with both absorption and scattering has been developed in the frame work of Discrete Space Theory. Two cases have been considered: (A) irradiation of the atmosphere at =T and (B) no irradiation on either side of the atmosphere. Isotropic scattering has been assumed. It is found that the emergent luminosities (defined by r 2 I(r, ) ) from scattering dominated atmospheres are smaller than those from absorption dominated atmospheres.  相似文献   

16.
For the evaluation of the effect of the non-uniform surface albedo on the emergent radiation from the atmosphere, the emergent radiation from the atmosphere bounded by the two half Lambert surfaces composed of different albedo is computed. This paper is the improved version of the previous paper (Takashima and Masuda, 1991). The atmosphere is assumed to be homogeneous, which is composed of aerosol, molecules, and absorbent gases. Their optical thicknesses are (1) 0.25, 0.23, and 0.02, and (2) 0.75, 0.23, and 0.02, respectively. The model aerosol is of the oceanic and water soluble types.In the computational procedure, the emergent radiation is calculated approximately by the contributions due to the multiple scattering in the atmosphere, and due to the diffusely or directly transmitted radiation through the atmosphere which is reflected by the surfaces once (4 interactive radiative modes between atmosphere and surface). Furthermore, to perform the hemispherical integration processing the radiative interaction, the transmission function based on the single scattering in the atmosphere is introduced and then the transmission function is averaged over the hemisphere with weighting function. The numerical simulation exhibits the extraordinary effect near the two half surface boundary of different albedoes. The effect decreases exponentially with the distance from the boundary. The effect depends on the atmospheric aerosol type, optical thickness, and surface albedo. The present version enables us to quantitatively discuss the radiative transfer trend near the boundary of two half surfaces. The upward radiance would simply be evaluated using the present scattering approximation method if the surface albedo is less than 0.3. The present method is thought of as a first step extending the one-dimensional radiative transfer model to two-dimensional using the doubling-adding method.  相似文献   

17.
The Fredholm integral equation method (FIM), originally introduced by Holtet al. to solve the light scattering problem for ellipsoidal particles, is reinvestigated by taking into account a recent great progress in numerical computers. A numerical code optimized for vector-processing computers is developed, and is applied to the light scattering by spherical and spheroidal particles. The results for these particles are compared with those by the Mie theory and by Asano and Yamamoto, respectively, and it is confirmed that the agreement with both of them is satisfactory. Sample calculations are also performed for the oblique incidence, in which the direction of incidence is not parallel nor perpendicular to the symmetry axis of the particle. No difficulties in the computation are found compared with the calculations for the parallel or perpendicular incidence. We study the efficiency factor for polarization (Q pol) in general direction of incidence for spheroidal particles, and discuss the deviation from the Rayleigh approximation.  相似文献   

18.
Transneptunian objects (TNOs) orbit beyond Neptune and do offer important clues about the formation of our solar system. Although observations have been increasing the number of discovered TNOs and improving their orbital elements, very little is known about elementary physical properties such as sizes, albedos and compositions. Due to TNOs large distances (>40 AU) and observational limitations, reliable physical information can be obtained only from brighter objects (supposedly larger bodies). According to size and albedo measurements available, it is evident the traditionally assumed albedo p=0.04 cannot hold for all TNOs, especially those with approximately absolute magnitudes H?5.5. That is, the largest TNOs possess higher albedos (generally >0.04) that strongly appear to increase as a function of size. Using a compilation of published data, we derived empirical relations which can provide estimations of diameters and albedos as a function of absolute magnitude. Calculations result in more accurate size/albedo estimations for TNOs with H?5.5 than just assuming p=0.04. Nevertheless, considering low statistics, the value p=0.04 sounds still convenient for H>5.5 non-binary TNOs as a group. We also discuss about physical processes (e.g., collisions, intrinsic activity and the presence of tenuous atmospheres) responsible for the increase of albedo among large bodies. Currently, all big TNOs (>700 km) would be capable to sustain thin atmospheres or icy frosts composed of CH4, CO or N2 even for body bulk densities as low as 0.5 g cm−3. A size-dependent albedo has important consequences for the TNOs size distribution, cumulative luminosity function and total mass estimations. According to our analysis, the latter can be reduced up to 50% if higher albedos are common among large bodies.Lastly, by analyzing orbital properties of classical TNOs (), we confirm that cold and hot classical TNOs have different concentration of large bodies. For both populations, distinct absolute magnitude distributions are maximized for an inclination threshold equal to 4.5° at >99.63% confidence level. Furthermore, more massive classical bodies are anomalously present at , a result statistically significant and apparently not caused by observational biases. This feature would provide a new constraint for transneptunian belt formation models.  相似文献   

19.
Simultaneous observations of the polarization of radiations H and D3 have been performed in 1979 at Pic du Midi on 60 solar prominences (300 pairs of measurements). For the essential part of the sample, the degree of polarization p(D3) does not depend strongly upon the intensity E(D3); on the contrary p(H) decreases steeply when E(H) goes beyond the brightness which corresponds to (H) = 1 (Figure 4); the deviation of the polarization direction (H) does not show such a strong dependance upon E(H) (Figure 6). A crude estimate of the possible role of multiple scattering in an optically thick prominence is in general agreement with observational data but a detailed comparison will be possible only when a complete theory of the Hanle effect in non optically thin layers will be available.Therefore, H polarization measurements are presently useful mainly when (H) < 1 and, in connexion with D3 measurements (Figure 8), they should make possible to determine the three components of the prominence magnetic vector. Prospects are given for the observation of other lines (H; 10 830 Å) which do not suffer heavy self-absorption effects and would be well-suited to this type of study.  相似文献   

20.
Michael J. Price 《Icarus》1977,30(4):760-768
Visual photometric function data for Saturn's rings are analyzed in terms of elementary anisotropic scattering radiative transfer models which involve the Henyey-Greenstein function. Limits are placed on the combinations of single scattering albedo and backscattering directivity which are permitted by observation. Particles with both microscopic and macrscopic lunar-like scattering properties are excluded by the analysis. Results are consistent with the ring particles being nearly pure spherical conglomerates of H2O frost.  相似文献   

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