Ambartsumian’s methods in the theory of radiative transfer

The purpose of this article is to provide some insight into Ambartsumian’s methods in the theory of radiative transfer, their applications, and further development. Two of these methods are emphasized--the invariance principle and the method of addition of layers, proposed by Ambartsumian in the 1940’s. The difference between these methods and the classical approach for solving radiative transfer problems is discussed. We discuss only a small portion of the subsequent work by others that we believe reveals, in a more intuitive way, the essence and significance of Ambartsumian’s methods and their efficiency for applications. Thus, for example, a separate section is devoted to applications of the Lagrangian formalism to radiative transfer and it is shown that the invariance principle is a special case of a more general variational principle that reflects an invariance with respect to translational transformation of the optical depth. Our discussion of the method of addition of layers points out its generality and the major role it has played in the later creation of such methods as Bellman’s invariant imbedding method and the method for solving radiative transfer problems in inhomogeneous media. The latter method has yielded a number of new analytic results. The concluding section is a brief summary of Ambartsumian’s results in the nonlinear theory of radiative transfer, where he was a pioneer in the study of the class of multilevel problems. This article also sets out to demonstrate the place and role of Ambartsumian’s methods in the theory of radiative transfer, which, to a great extent, set the path along which this theory developed for many years to come. Translated from Astrofizika, Vol. 52, No. 1, pp. 5–27 (February 2009).     

Optimal low-thrust transfers between libration point orbits

Over the past three decades, ballistic and impulsive trajectories between libration point orbits (LPOs) in the Sun–Earth–Moon system have been investigated to a large extent. It is known that coupling invariant manifolds of LPOs of two different circular restricted three-body problems (i.e., the Sun–Earth and the Earth–Moon systems) can lead to significant mass savings in specific transfers, such as from a low Earth orbit to the Moon’s vicinity. Previous investigations on this issue mainly considered the use of impulsive maneuvers along the trajectory. Here we investigate the dynamical effects of replacing impulsive ΔV’s with low-thrust trajectory arcs to connect LPOs using invariant manifold dynamics. Our investigation shows that the use of low-thrust propulsion in a particular phase of the transfer and the adoption of a more realistic Sun–Earth–Moon four-body model can provide better and more propellant-efficient solution. For this purpose, methods have been developed to compute the invariant tori and their manifolds in this dynamical model.     

Indirect transfer to the Earth–Moon L1 libration point

The Earth–Moon L1 libration point is proposed as a human gateway for space transportation system of the future. This paper studies indirect transfer using the perturbed stable manifold and lunar flyby to the Earth–Moon L1 libration point. Although traditional studies indicate that indirect transfer to the Earth–Moon L1 libration point does not save much fuel, this study shows that energy efficient indirect transfer using the perturbed stable manifold and lunar flyby could be constructed in an elegant way. The design process is given to construct indirect transfer to the Earth–Moon L1 libration point. Simulation results show that indirect transfer to the Earth–Moon L1 libration point saves about 420 m/s maneuver velocity compared to direct transfer, although the flight time is about 20 days longer.     

On quasi-periodic motions around the collinear libration points in the real Earth–Moon system

Due to various perturbations, the collinear libration points of the real Earth–Moon system are not equilibrium points anymore. Under the assumption that the Moon’s motion is quasi-periodic, special quasi-periodic orbits called dynamical substitutes exist. These dynamical substitutes replace the geometrical collinear libration points as time-varying equilibrium points. In the paper, the dynamical substitutes of the three collinear libration points in the real Earth–Moon system are computed. For the points L ₁ and L ₂, linearized motions around the dynamical substitutes are described, and the variational equations of the dynamical substitutes are reduced to a form with a near constant coefficient matrix. Then higher order analytical formulae of the central manifolds are constructed. Using these analytical solutions as initial seeds, Lissajous orbits and halo orbits are computed with numerical algorithms.     

Two‐Body Problems with Drag or Thrust: Qualitative Results

We consider two‐body problems in which the drag is proportional to the velocity divided by the square of the distance and whose radial and tangential components have distinct coefficients. For all parameters, we study the flow of the system obtained by suitable coordinate and time transformations and draw conclusions about the qualitative behavior of solutions. In each case, we examine the existence of collision–ejection, collision–escape, capture–collision, capture–escape, and oscillatory rectilinear orbits, study the motion near collision, and show that if periodic orbits exist they must be limit cycles. This revised version was published online in July 2006 with corrections to the Cover Date.     

On derivation of EIH (Einstein–Infeld–Hoffman) equations of motion from the linearized metric of general relativity theory

The half-century old idea of Infeld to use the variational principle of the general relativity field equations is reminded to show that the commonly employed EIH (Einstein–Infeld–Hoffman) equations of motion may be derived from the linearized (weak-field) metric alone. Based on it, the linearized metric might be sufficient for post-Newtonian celestial mechanics and astrometry enabling one to derive the post-Newtonian equations of motion and rotation of celestial bodies as well as the post-Newtonian equations of light propagation within the general relativity framework.     

The Shadow of Comet Hale–Bopp in Lyman-Alpha

For a few months around perihelion, thecentral part of the Hale–Bopp hydrogencloud has been optically thick to thesolar Lyα radiation, and hassignificantly reduced the solar flux availablefor the resonance glow of interstellarhydrogen beyond the comet. This shadowing effecton the interstellar gas is the first everobserved comet shadow. It is modeled andcompared with SWAN observations. Shadowmodelling will help to constrain the cometwater production and radiative transfer effectsin the interstellar ionisation cavity.     

Solution of linear radiative transport problems in plane-parallel atmospheres. I.

A new method for determining various quantities describing the radiation field in an inhomogeneous, plane-parallel atmosphere is proposed in this two-part paper. The essence of this method is the reduction of the boundary value problems which arise during the customary statement of various astrophysical problems associated with solving the radiative transfer equations to initial value problems. Compared to previous attempts in this area, the proposed method is universal and simple. The first part of this paper deals with one-dimensional media. Scalar, as well as vector–matrix problems relating to the diffusion of radiation in spectral lines with frequency redistribution are examined.     

Extension of fast periodic transfer orbits from the Earth–Moon RTBP to the Sun–Earth–Moon Quasi-Bicircular Problem

Starting from 80 families of low-energy fast periodic transfer orbits in the Earth–Moon planar circular Restricted Three Body Problem (RTBP), we obtain by analytical continuation 11 periodic orbits and 25 periodic arcs with similar properties in the Sun–Earth–Moon Quasi-Bicircular Problem (QBCP). A novel and very simple procedure is introduced giving the solar phases at which to attempt continuation. Detailed numerical results for each periodic orbit and arc found are given, including their stability parameters and minimal distances to the Earth and Moon. The periods of these orbits are between 2.5 and 5 synodic months, their energies are among the lowest possible to achieve an Earth–Moon transfer, and they show a diversity of circumlunar trajectories, making them good candidates for missions requiring repeated passages around the Earth and the Moon with close approaches to the last.     

Ambartsumian’s invariance principle and some nonlinear relations in radiative transfer theory

This paper is devoted to one of the methods proposed by Ambartsumian in radiative transfer theory— the invariance principle. The possible connection of several well known nonlinear relations in the theory to a variational principle involving a translational transformation of the optical depth is discussed.     

Variation of the Solar Neutrino

The investigation of the solar neutrinos using the experimental data base of the Davis Cl-Ar experiment for a more than 20-year period shows that two problems take place. First – registered neutrino flux was three times smaller than predicted; second is variations of the solar neutrinos. For the last few years a number of papers have appeared in which both the theoretical and experimental data are presented and it is known that extremely contradictory results take place. The present paper is devoted to the second problem of the solar neutrinos – their variations. This revised version was published online in July 2006 with corrections to the Cover Date.     

Prompt determination of universal time corrections in e-VLBI mode

The results of a prompt determination of the Universal Time corrections in the e-VLBI mode on the Quasar VLBI network in 2009–2011 are presented. For this purpose, the hardware-software tools for observational data transfer from the Svetloe, Zelenchukskaya, and Badary Observatories to the Correlation Processing Center (St. Petersburg) over fiber-optic communication lines in a mode close to real time using the specialized Tsunami protocol have been developed. The rms error in the UTC-UT1 corrections with respect to the combined IERS series is 60 μs.     

Fast Periodic Transfer Orbits in the Sun–Earth–Moon Quasi-Bicircular Problem

Starting from the identification and classification of a family of fast periodic transfer orbits in the Earth–Moon planar circular Restricted Three Body Problem (RTBP), and using analytic continuation techniques, we find two unstable periodic orbits in the Sun–Earth–Moon Quasi-Bicircular Problem (QBCP). The orbits found perform periodic Earth–Moon transfers with a period of approximately 29.5 days.     

Nonlinear equations for statistical averages describing processes of multiple scattering

The Lagrangian formalism is used to obtain nonlinear equations for frequently encountered statistical characteristics of radiative transfer. Under the assumption of complete frequency redistribution, the average number of scatterings and the average time of stay of a photon in a one-dimensional, semi-infinite atmosphere are analyzed. Two typical problems of multiple scattering are solved to illustrate the possible applications of the equations obtained. Translated from Astrofizika, Vol. 43, No. 3, pp. 463-471, July– September, 2000.     

The experiment with the fast X-ray monitor (BRM) instrument onboard the CORONAS-PHOTON satellite

The “Fast X-ray Monitor” (BRM) instrument operated in the complex of the scientific instruments onboard the CORONAS-PHOTON satellite from February 19, 2009, until December 1, 2009. The instrument is intended for the registration of the hard X-ray radiation of solar flares in the 20–600 keV energy range in six differential energy channels (20–30, 30–40, 40–50, 50–70, 70–130, and 130–600 keV) with temporal resolution to 1 ms. In the instrument, a detector based on the YAP: Ce scintillator is used; this detector is 70 mm in diameter and 10 mm thick (the decay time is about 28 ns). For the decrease of the back-ground charge of the detector, the collimator limiting the angle of view of the instrument of value 12° is mounted over the scintillator. The effective area of the detector amounts to 27.7 cm² (at the X-ray radiation energy 80 keV), and the dead time of the detector is 1 μs. Over the operation onboard the CORONAS-PHOTON satellite, the BRM instrument has registered gamma ray burst series and, perhaps, one solar flare of the class C1.3 on October 26, 2009.     

Optimal transfers between unstable periodic orbits using invariant manifolds

This paper presents a method to construct optimal transfers between unstable periodic orbits of differing energies using invariant manifolds. The transfers constructed in this method asymptotically depart the initial orbit on a trajectory contained within the unstable manifold of the initial orbit and later, asymptotically arrive at the final orbit on a trajectory contained within the stable manifold of the final orbit. Primer vector theory is applied to a transfer to determine the optimal maneuvers required to create the bridging trajectory that connects the unstable and stable manifold trajectories. Transfers are constructed between unstable periodic orbits in the Sun–Earth, Earth–Moon, and Jupiter-Europa three-body systems. Multiple solutions are found between the same initial and final orbits, where certain solutions retrace interior portions of the trajectory. All transfers created satisfy the conditions for optimality. The costs of transfers constructed using manifolds are compared to the costs of transfers constructed without the use of manifolds. In all cases, the total cost of the transfer is significantly lower when invariant manifolds are used in the transfer construction. In many cases, the transfers that employ invariant manifolds are three times more efficient, in terms of fuel expenditure, than the transfer that do not. The decrease in transfer cost is accompanied by an increase in transfer time of flight.     

Albedo-shift method in the problem of anisotropic light scattering in a plane atmosphere

A standard problem of radiative transfer theory — calculating the diffuse reflection and transmission of radiation by a plane scattering atmosphere — is considered. The recently proposed albedoshift method is used to calculate the X and Y functions (and the H function) for the case of anisotropic scattering with a Henyey-Greenstein indicatrix. The method enables one to “suppress” scattering and obtain iterative solutions of high accuracy in only a few iterations, even when the mean number of photon scatterings in the atmosphere is very large. Translated from Astrofizika, Vol. 41. No. 4, pp. 623–646, October–December, 1998.     

Albedo-shifting method: Source function in a plane atmosphere

A classical problem in the theory of radiative transfer is considered: calculating the radiation field within a plane scattering atmosphere. The recently proposed albedo-shifting method is used to calculate the source function both in a semi-infinite atmosphere and in an atmophere of finite optical depth, illuminated by parallel rays. The method enables one to “suppress” scattering and obtain iterative solutions of the integral equation for the source function in only a few direct lambda iterations, even when the average number of photon scatterings in the atmosphere is very large. Translated from Astrofizika, Vol. 42, No. 4, pp. 485–500, October–December, 1999.     

Instability and Diffusion in the Elliptic Restricted Three-body Problem

The importance of the stability characteristics of the planar elliptic restricted three-body problem is that they offer insight about the general dynamical mechanisms causing instability in celestial mechanics. To analyze these concerns, elliptic–elliptic and hyperbolic–elliptic resonance orbits (periodic solutions with lower period) are numerically discovered by use of Newton's differential correction method. We find indications of stability for the elliptic–elliptic resonance orbits because slightly perturbed orbits define a corresponding two-dimensional invariant manifold on the Poincaré surface-section. For the resonance orbit of the hyperbolic–elliptic type, we show numerically that its stable and unstable manifolds intersect transversally in phase-space to induce instability. Then, we find indications that there are orbits which jump from one resonance zone to the next before escaping to infinity. This phenomenon is related to the so-called Arnold diffusion. This revised version was published online in August 2006 with corrections to the Cover Date.     

Analysis of Sunspot Activity Cycles

A nonlinear analysis of the daily sunspot number for each of cycles 10 to 23 is used to indicate whether the convective turbulence is stochastic or chaotic. There is a short review of recent papers considering sunspot statistics and solar activity cycles. The differences in the three possible regimes – deterministic laminar flow, chaotic flow, and stochastic flow – are discussed. The length of data sets necessary to analyze the regimes is investigated. Chaos is described and a chronology of recent results that utilize chaos and fractals to analyze sunspot numbers follows. The parameters necessary to describe chaos – time lag, phase space, embedding dimension, local dimension, correlation dimension, and the Lyapunov exponents – are determined for the attractor for each cycle. Assuming the laminar regime is unlikely if chaos is not indicated in a cycle by the calculations, the regime must be stochastic. The sunspot numbers in each of cycles 10 to 19 indicate stochastic behavior. There is a transition from stochastic to chaotic behavior of the sunspot numbers in cycles 20, 21, 22, and 23. These changes in cycles 20 – 23 may indicate a change in the scale of turbulence in the convection zone that could result in a change in the convective heat transfer and a change in the size of the convection region for these four cycles.