An improved version of the implicit integral method to solving radiative transfer problems

Radiative transfer (RT) problems in which the source function includes a scattering-like integral are typical two-points boundary problems. Their solution via differential equations implies making hypotheses on the solution itself, namely the specific intensity I (τ; n) of the radiation field. On the contrary, integral methods require making hypotheses on the source function S(τ). It seems of course more reasonable to make hypotheses on the latter because one can expect that the run of S(τ) with depth is smoother than that of I (τ; n). In previous works we assumed a piecewise parabolic approximation for the source function, which warrants the continuity of S(τ) and its first derivative at each depth point. Here we impose the continuity of the second derivative S′′(τ). In other words, we adopt a cubic spline representation to the source function, which highly stabilizes the numerical processes.     

A new method for the statistical simulation of the virtual values of parameters in inverse orbital dynamics problems

A Monte Carlo-type method for simulating virtual values of the parameters in inverse orbital dynamics problems for highly nonlinear cases is proposed. The method is based on imitating Fisher’s statistics employed to specify the confidence region, and is implemented by solving repeatedly nonlinear least-squares problems with various samples of simulated observations obtainable by suitable random variations.     

A new exact method for line radiative transfer

We present a new method, the coupled escape probability (CEP), for exact calculation of line emission from multi-level systems, solving only algebraic equations for the level populations. The CEP formulation of the classical two-level problem is a set of linear equations , and we uncover an exact analytic expression for the emission from two-level optically thick sources that holds as long as they are in the 'effectively thin' regime. In a comparative study of a number of standard problems, the CEP method outperformed the leading line transfer methods by substantial margins.
The algebraic equations employed by our new method are already incorporated in numerous codes based on the escape probability approximation. All that is required for an exact solution with these existing codes is to augment the expression for the escape probability with simple zone-coupling terms. As an application, we find that standard escape probability calculations generally produce the correct cooling emission by the C  ii 158-μm line but not by the ³P lines of O  i .     

TheF N method for solving radiative-transfer problems in plane geometry

TheF _N method of solving problems in radiative transfer for plane-parallel media with anisotropic scattering is established. The method utilizes properties of the exact solution and leads to final equations that are particularly concise and easy to use.     

A new numerical method for solving radiation driven winds from hot stars

We present a general method for solving the non‐linear differential equation of monotonically increasing steady‐state radiation driven winds. We graphically identify all the singular points before transforming the momentum equation to a system of differential equations with all the gradients explicitly given. This permits a topological classification of all singular points and to calculate the maximum and minimum mass‐loss of the wind. We use our method to analyse for the first time the topology of the non‐rotating frozen‐in ionisation m‐CAK wind, with the inclusion of the finite disk correction factor, and find up to 4 singular points, three of the x‐type and one attractor‐type. The only singular point (and solution passing through) that satisfies the boundary condition at the stellar surface is the standard m‐CAK singular point. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A new moment method for continuum radiative transfer in cosmological re-ionization

We introduce a new code for computing time-dependent continuum radiative transfer and non-equilibrium ionization states in static density fields with periodic boundaries. Our code solves the moments of the radiative transfer equation, closed by an Eddington tensor computed using a long characteristics (LC) method. We show that traditional short characteristics and the optically thin approximation are inappropriate for computing Eddington factors for the problem of cosmological re-ionization. We evolve the non-equilibrium ionization field via an efficient and accurate (errors <1 per cent) technique that switches between fully implicit or explicit finite differencing depending on whether the local time-scales are long or short compared to the time-step. We tailor our code for the problem of cosmological re-ionization. In tests, the code conserves photons, accurately treats cosmological effects and reproduces analytic Strömgren sphere solutions. Its chief weakness is that the computation time for the LC calculation scales relatively poorly compared to other techniques  ( t _LC∝ N ^∼1.5_cells)  ; however, we mitigate this by only recomputing the Eddington tensor when the radiation field changes substantially. Our technique makes almost no physical approximations, so it provides a way to benchmark faster but more approximate techniques. It can readily be extended to evolve multiple frequencies, though we do not do so here. Finally, we note that our method is generally applicable to any problem involving the transfer of continuum radiation through a periodic volume.     

A new method for exact solution of transfer equations in finite media

In this paper we develop a new exact method combined with finite Laplace transform and theory of linear singular operators to obtain a solution of transport equation in finite plane-parallel steady-state scattering atmosphere both for angular distribution of radiation from the bounding faces of the atmosphere and for intensity of radiation at any depth of the atmosphere. The emergent intensity of radiation from the bounding faces are determined from simultaneous linear integral equations of the emergent intensity of radiation in terms ofX andY equations of Chandrasekhar. The intensity of radiation at any optical depth for a positive and negative direction parameter is derived by inversion of the Laplace transform in terms of intergrals of the emergent intensity of radiation. A new expression of theX andY equation is also derived for easy numerical computation. This is a new and exact method applicable to all problems in finite plane parallel steady scattering atmosphere.     

A note on modified spherical-harmonic method for radiative transfer problems with reflective boundary conditions

The modified double-interval spherical-harmonic method is used to compute the radiative flux in a linearly anisotropically scattering plane-parallel medium with specularly and diffusely reflecting boundaries.     

Generalized Lagrangian orbital elements for central force problems

We consider the definitions and resulting equations of motion for the Lagrangian orbital elements associated with conventional osculating orbit theory for central forces. The analysis indicates that the definitions themselves lead to difficulties which are most apparent in the circular limit. An alternate set of defining relations is presented which eliminates the problems associated with osculating elements. The remaining equation of motion based on these new definitions is reduced to quadratures. This solution completely expresses the orbits for central force problems with no restriction on the eccentricity. Both bounded and open orbits are considered. A generalized Laplace-Runge-Lenz vector is developed and a number of example solutions are presented.     

A sixth-order accurate scheme for solving two-point boundary value problems in astrodynamics

A sixth-order accurate scheme is presented for the solution of ODE systems supplemented by two-point boundary conditions. The proposed integration scheme is a linear multi-point method of sixth-order accuracy successfully used in fluid dynamics and implemented for the first time in astrodynamics applications. A discretization molecule made up of just four grid points attains a O(h ⁶) accuracy which is beyond the first Dahlquist’s stability barrier. Astrodynamics applications concern the computation of libration point halo orbits, in the restricted three- and four-body models, and the design of an optimal control strategy for a low thrust libration point mission.     

An intermediate matching technique for solving two point boundary value problems using the perturbation method

The perturbation method, a numerical method for solving two point boundary value problems (TPBVP), is modified to attempt to improve inherent instability and sensitivity problems associated with the method. The desired solution to the TPBVP is divided into two time intervals. The differential equations required to define a solution to the two point boundary value problem are integrated independently over these shorter segments rather than consecutively over the entire trajectory. The independent integration of the differential equations over approximately half of the trajectory instead of the entire trajectory substantially decreases sensitivity and stability properties associated with the numerical integration. The equations for both time segments can be integrated simultaneously. By this procedure, a system of twice the dimension of the original problem is integrated for a period of time equal to half of the time interval for the original problem. To show the effectiveness of the method, two impulse trajectories which minimize the total velocity increment required to transfer a spacecraft from an Earth orbit into a lunar orbit are calculated.     

On combined operations method for transfer problems in homogeneous,cylindrical media

The problem of diffuse reflection by a homogeneous, isotropically scattering, infinite cylindrical medium has been considered. The relevant auxiliary equation has been formulated, the scattering function defined and the integro-differential equation for such function deduced. For a medium having cylindrical distribution of source in addition to the incident flux at the outer surface, the integro-differential equation for the emergent intensity has been established.     

On combined operational method for transfer problems in homogeneous,spherical media

In this paper, the Combined Operational Method developed by Busbridge (1961) in connection with the radiative transfer problems in plane-parallel atmospheres has been extended to similar problems in isotropic scattering, homogeneous spherical media. The relevant auxiliary equation has been formulated, the scattering function defined and the integro-differential equation for such function deduced. For a medium having radial distribution of source in addition to the incident flux at the outer surface, the integro-differential equations for source function and emergent intensity have been established.     

A special perturbation method in orbital dynamics

The special perturbation method considered in this paper combines simplicity of computer implementation, speed and precision, and can propagate the orbit of any material particle. The paper describes the evolution of some orbital elements based in Euler parameters, which are constants in the unperturbed problem, but which evolve in the time scale imposed by the perturbation. The variation of parameters technique is used to develop expressions for the derivatives of seven elements for the general case, which includes any type of perturbation. These basic differential equations are slightly modified by introducing one additional equation for the time, reaching a total order of eight. The method was developed in the Grupo de Dinámica de Tethers (GDT) of the UPM, as a tool for dynamic simulations of tethers. However, it can be used in any other field and with any kind of orbit and perturbation. It is free of singularities related to small inclination and/or eccentricity. The use of Euler parameters makes it robust. The perturbation forces are handled in a very simple way: the method requires their components in the orbital frame or in an inertial frame. A comparison with other schemes is performed in the paper to show the good performance of the method.     

Sinc-Collocation method for solving astrophysics equations

In this paper we propose Sinc-Collocation method for solving Lane–Emden equation which is a nonlinear ordinary differential equation on a semi-infinite interval. It is found that Sinc procedure converges with the solution at an exponential rate. This method is utilized to reduce the computation of this problem to some algebraic equations. We also compare this solution with some well-known results and show that it is accurate.     

Variational formulae for solving perturbed boundary-value problems in astrodynamics

The uniform valid state transition matrix of the pure Keplerian motion is established by means of the KS-transformation. Perturbed interplanetary transfer- and flyby-problems are treated by integrating the perturbed trajectory numerically and by performing the differential correction technique using the analytical transition matrix of the unperturbed motion. Numerical examples show the efficiency of this procedure and the improvement of the computing time and accuracy in comparison with the shooting method.     

Optimization of the transfer trajectory of a low-thrust spacecraft for research of Jupiter using an Earth gravity-assist maneuver

The paper analyzes the possibility of the use of a gravity-assist maneuver for flight to Jupiter. The advantage of the Earth gravity-assist maneuver in comparison with the direct transfer in terms of reduction of amount of energy required per transfer is considered. Quantitative and qualitative evaluations of two transfer profiles are given.     

Conservation laws for multilevel transfer problems

We discuss the question whether the way of finding the conservation laws based on variational formalism is applicable to the multilevel problems of radiative transfer in a homogeneous atmosphere. For expository reasons, the simplest one-dimensional model case is considered. For the special three-level problem treated in the paper the Lagrangian approach allows one to derive not only the H- and K-integrals, but also the nonlinear integral which is an analog of the Q-integrals previously obtained for the classical transfer problems. It is shown that, in general, the constraints imposed by the variational principle on the symmetry properties of the transfer equations are too stringent to be satisfied. Translated from Astrofizika, Vol. 42, No. 2, pp. 235–252, April–June, 1999.