Sueo Ueno

NEW DERIVATION OF THE INTEGRO-DIFFERENTIAL EQUATIONS FOR CHANDRASEKHAR'S X AND Y FUNCTIONS.

The X and Y functions of radiative transfer satisfy a system of integro‐differential equations which form the basis of an effective numerical treatment. These integro‐differential equations are derived from the integral equation for the source function and the differential equation for the resolvent.

On the time-dependent principle of invariance in a semi-infinite medium

For a semi-infinite medium in which the particle mean free time is much greater than the mean survival time of a photon (e.g,, envelopes of novae, planetary nebulae, and cosmic clouds), a radiative transfer theory is derived that tekes into account the irreversible leakage of radiation from the system. The time-dependent diffusion problem posed by this theory is solved by means of a transient scattering function. The explicit form of the integral equation for this scattering function is obtained. (T,F.H.)

Diffuse Transmission of Light from a Central Source through an Inhomogeneous Spherical Shell with Isotropic Scattering

A partial‐differential‐integral equation is derived in this paper for the angular distribution of the radiation which is diffusely transmitted through an inhomogeneous, isotropically scattering, spherical shell when there is a constant net flux of radiation normally incident on the inner surface. An equation is also derived for the strength of the diffusely reflected radiation when the shell is illuminated at each point on the outer surface by constant isotropic incident radiation.The equations obtained appear to lend themselves well to numerical solution. Astrophysically, the situation corresponds to determining the brightness of a spherical planetary nebula. As far as is known, the equations are new and exact.

Chandrasekhar's planetary problem with internal sources

Abstract It is of interest to try to determine the vertical stratification of terrestrial and planetary atmospheres based on observations made from artificial satellites. For example, the altitude of the upper boundary of clouds may be estimated from the relative spectral intensities of the solar radiation reflected from the cloud in the oxygen “A” band at 7600 A and in a nearby band. This paper deals with the mathematical treatment of Chandrasekhars planetary problem in anisotropically scattering, inhomogeneous finite atmospheres which contain emitting sources of radiation. The initial-value problems for differential-integral equations obtained via the invariant imbedding approach are suitable for numerical solution. The same equations will be useful in the computational treatment of inverse problems for the determination of cloud heights and other parameters.

The probabilistic method for problems of radiative transfer XIII. Diffusion matrix

In the theory of radiative transfer the equation of transfer formulated from the local viewpoint is expressed in terms of two optical parameters, i.e. an albedo for single scattering and an indicatrix of scattering. These optical parameters are assumed classically to be constant throughout the layer under consideration. In the present paper we restrict our discussion to the exact solutions of stationary transfer problems of radiation. The Laplace transform method due to Wiener and Hopf [l] and the Chandrasekhar limiting process [2] are considered to be the local approach for solving exactly the equation of transfer. The above two methods have further been extended by Miss Busbridge [3, 41, Huang [5], and Kourganoff [3], respectively. Reducing the Milne first integral equation to the auxiliary equation by the idea of linear aggregation, Ambarzumian [l] has elegantly developed a new method giving the exact solution of the transfer equation. Then the extension of Ambarzumian’s first method has been done in several transfer problems of current interest in astrophysics: line formation in semi-infinite, coherently and non-coherently scattering atmosphere (Busbridge [4] ; Ueno [6]), and line formation in a coherently scattering atmosphere of finite thickness (Busbridge [7, 41). Furthermore, in 1943 Ambarzumian [8] initiated a new method based on the principle of invariance. The method is considered to be the global approach, because a nonlinear integral equation in +-function can directly be obtained by physical analysis of the condition of transfer without actually solving the Milne first integral equation for the source function on the way. The extension of Ambarzumian’s physical method has been made by Miss

A probabilistic approach for scattering of light in slab geometry—I

Abstract : A new probabilistic approach to radiative transfer problems is presented in such a way that the integral equations for the stationary and nonstationary scattering functions in a finite inhomogeneous flat layer are derived directly from a somewhat modified form of the Chapman-Kolmogoroff equation.

SOURCE FUNCTIONS FOR AN ISOTROPICALLY SCATTERING ATMOSPHERE BOUNDED BY A SPECULAR REFLECTOR

Abstract It is desired to calculate source functions for an isotropically scattering finitely thick and absorbing atmosphere which is bounded by a specular reflector and illuminated by parallel rays. These source functions satisfy a basic Fredholm integral equation. This paper presents a new method for the solution of such integral equations. It is well suited for numerical solution, and the results of extensive calculations are presented. The method extends to systems of integral equations, such as are encountered when the effects of polarization are included. These systems are currently under study.

The invariant imbedding method for transport problems II. Resolvent in photon diffusion equation

In recent years, making use of functional equation techniques in various ways in connection with the principles of inrariance due to Ambanumian [l] and Chandrasekhar [15], the study of radiative transfer has exactly been carried out by Sobolev [al, 221, Miss Busbridge [lo, 111, Horak and Lundquist [IS], Stibbs [ia, 251, Bellman and Kalaba [2, 32, King [17], Preisendorfer [19, 201, and Ueno [SS, 291, respectively. Let a particular physical process be given. Then, by imbedding this process within an appropriate class of processes the functional relationships existing among the various processes of the class will be found. In such a manner the reflected and transmitted fluxes and the probability distributions for these fluxes in neutron transport theory have been computed by Bellman, Kalaba, and Wing [4,9]. Recently, allowing for the probability significance of the resolvent in the Milne first integral equation for the source function j(t)

On the principle of invariance in a semi-infinite inhomogeneous atmosphere.

With the aid of the extension of Chandrasekhars invariance method and the principle of invariant imbedding stated by Bellman and Kalaba, respectively, we obtain the emergent intensity of diffusely reflected radiation in a semi-infinite atmosphere of arbitrary stratification The result reduces to that given by Sobolev and later by us.

System identification: Associate memories for system identification: Inverse problems in remote sensing

The problems of remote sensing of physical parameters from satellites is an important inverse, or system identification problem. Progress has been made in several phases. During the 1960s basic equations of radiative transfer in various types of media (inhomogeneous, emitting, anisotropic, polarized, etc.) were derived for the treatment of direct problems, i.e., problems in which the parameters were known. In order to avoid instabilities in numerically obtaining solutions, nonlinear Cauchy problems for systems of ordinary differential equations were developed. This formulation aided in the treatment of inverse problems, where the parameters are to be estimated based on observations of external or internal radiant energy. System identification techniques such as quasilinearization were found to be effective, when good preliminary estimates were available. Recently, a systematic procedure for obtaining the needed initial estimates has been investigated. This approach, based on the concept of associative memories, has already produced excellent computational results.