A nonstationary transfer problem with partial frequency redistribution

We analyze a system of equations for nonstationary transfer of resonant radiation taking into account nonlinear effects, and with partial frequency redistribution using a two-level model of the atom. We suggest a method for solving the corresponding problem in a three-dimensional plane-parallel medium.Translated from Astrofizika, Vol. 38, No. 1, pp. 87–97, January–March, 1995.     

The problem of stellar membership in open clusters

Open clusters become a powerful astrophysics laboratory once the essential problem of stellar membership has been solved. In this paper different methods to compute membership probabilities using kinematical variables, mainly proper motions, are presented.Presented at the 2nd UN/ESA Workshop, held in Bogotá, Colombia, 9–13 November, 1992.     

Inverse problem for determining the potential and velocity field in the galactic plane

Leningrad State University. Translated from Astrofizika, Vol. 30, No. 3, pp. 643–653, May–June, 1989.     

Nonlinear problem of polychromatic scattering;

The paper deals with a nonlinear transfer problem of polychromatic scattering in case of general laws of the frequency redistribution in a three-dimensional medium consisting of atoms with three discrete energy levels. Transitions 2–3 are assumed to be completely forbidden. Application of Ambarzumian's method of self-consistent optical depths allows us to linearize the problem. Results of numerical calculations are given for the case of the Doppler redistribution law. Numerical results are applied to the sodium resonant doublet.     

Evolution of low-mass close binaries and the problem of orbital period gap in cataclysmic variables

Evolution of close binary composed of a white dwarf primary and a Main-Sequence secondary has been calculated. Angular-momentum loss via gravitational radiation and magnetic stellar wind have been taken into account. We have found that magnetic stellar wind with a rate greater than (10^–10–10^–9)M yr^–1 is able to drive the evolution with mass exchange. If the time-scale of switch-off of wind when the primary becomes fully convective is not longer than 10⁶ yr, mass exchange interrupts due to a contraction of the secondary and the system becomes unobservable. Mass exchange resumes when components approach one another due to loss of momentum via gravitational radiation. The location and width of the thus-arising gap in the orbital periods are comparable to those observed.     

The existence of families of periodic orbits in theN-body problem

It is proved that monoparametric families of periodic orbits of theN-body problem in the plane, for fixed values of all masses, exist in a rotating frame of reference whosex axis contains always two of the bodiesP ₁ andP ₂. A periodic motion of theN-body problem is obtained as a continuation ofN–2 symmetric periodic orbits of the circular restricted three-body problem whose periods are in integer dependence, by increasing the masses of the originallyN–2 massless bodiesP ₃, ...,P _k. The analytic continuation, for sufficiently small values of theN–2 bodiesP ₃ ...P _k and finite values for the masses ofP ₁ andP ₂ has been proved by the continuation method and the solution itself has been found explicitly to a linear approximation in the small masses. Also, the possible application of the above periodic orbits to the study of the Solar system and of stellar systems is mentioned.     

Canonical solution of the two critical argument problem

The solution by Sessin and Ferraz-Mello (Celes. Mech. 32, 307–332) of the Hori auxiliary system for the motion of two planets with periods nearly commensurate in the ratio 21 is considerably simplified by the introduction of canonical variables. An analogous canonical transformation simplifies the elliptic restricted problem.     

Collective plasma effects and the electron number problem in solar hard X-ray bursts

Due to the relatively high stream densities involved, collective interactions with the ambient plasma are likely to be important for the electrons producing solar hard X-ray bursts. In thick- and thin-target bremsstrahlung models the most relevant process is limitation of the invoked electron beams by ion sound wave generation in the neutralizing reverse current established in the atmosphere. For the thick target model it is shown that typical electron fluxes are near the maximum permitted by stability of the reverse current so that ion-sound wave generation may be the process which limits the electron injection rate. On the other hand the chromospheric reverse current is sufficient to supply the large total number of electrons which have to be accelerated in the corona. For the thin target the low density of the corona severely limits the possible reverse current so that the maximum upward flux of fast electrons is probably much too small to explain X-ray bursts but compatible with observations of interplanetary electrons.A distinct class of model postulates a small number of electrons confined by resonant scattering in a dense coronal slab surrounding a current sheet with continuous stochastic acceleration offsetting collisional losses. The energetic aspects of such a situation described by Hoyng (1975) are developed here by addition of equations describing the slab geometry in terms of electron diffusion by whistler scattering and of the collisional damping of the accelerating Langmuir waves. Solution of these equations results in values for the fieldB(70–350 G), densityn ₀(2–5 × 10¹² cm ^–3), slab dimensions (10¹⁸ km² × 0.3–3 km) and relative Langmuir energy density (10^–3 – 10^–2) required to produce the observed range of bursts. It is pointed out, however, that there may be no real gain in electron number requirements since the fast electrons in the emitting slab would be constantly swept out along with the frozen-in plasma as dissipation proceeds so that a large total number of electrons is still required. It could in fact be that just such a coronal region is the injection mechanism for the thick-target model.On leave from Department of Astronomy, University of Glasgow, Scotland.     

The almost constant-speed two-body problem with resistance

The almost constant-speed motion of a mass acted upon by a Newtonian attraction and a resisting force is treated. The equation of orbit is derived for a specific type of resistance which covers the familiar case of Danby's drag(=r ^–2) whilst the vector invariants are obtained by direct operation on the vector form of the equation of motion.     

The rectilinear restricted problem of three bodies

In this paper we discuss some aspects of the isosceles case of the rectilinear restricted problem of three bodies, where two primaries of equal mass move on rectilinear ellipses, and the particle is confined to the symmetry axis of the system. In particular, the behaviour near a collision of the primaries and also near a collision of all three bodies is investigated. It is shown that this latter singularity is a triple collision in the sense of Siegel's theory. Furthermore, asymptotic expansions for the particle's motion during a parabolic and a hyperbolic escape are derived.Presented at the Conference on Celestial Mechanics, Oberwolfach, Germany, August 27–September 2, 1972.     

Solution of radiative transfer problem in a plane layer for the model of complete frequency redistribution. I. One-dimensional medium

Institute of Applied Problems of Physics, Armenian Academy of Sciences; Byurakan Astrophysical Observatory. Translated from Astrofizika, Vol. 28, No. 1, pp. 193–204, January–February, 1988.     

On the problem of the formation of quasar emission lines by the Cherenkov mechanism

In a series of papers of You, Cheng, et al. (cf., for example, [2–4]) the Cherenkov mechanism of radiation was proposed as the explanation of the wide emission lines in the spectra of quasars. This mechanism acts because the refractive index of the plasma in a certain frequency range near a spectral line can exceed one. It has been found that the role of Cherenkov radiation becomes significant for an optically thick medium when the density of the packet of relativistic electrons ñ_e is 10⁴–10⁶ cm^–3. However, the question of the formation of a spectral line as the result of ordinary (recombination and collision) processes under the same physical conditions has not been considered, and hence no comparison has been made between the intensities of lines formed by different mechanisms. Moreover, a large number of effects that may have significant influence on the profile of a line (multiple scattering in the medium, redistribution of radiation over frequencies within a line, redistribution of energy between a line and the continuous spectrum) have remained unexamined by these authors.The present work aims at sharpening some of the formulas used in [2–4] and filling up some of these gaps. We pose the classical problem of formation of a spectral line taking account of the action of the Cherenkov radiation mechanism and give an exact solution of it. The computations are carried out for the line L. It is shown that under the values presently accepted for the electron temperature for an average quasar (T_e 1.5 · 10⁴ K) the influence of the Cherenkov mechanism becomes noticeable only for implausibly high densities of relativistic electrons (10¹¹ cm^–3, The geometric model proposed in [4] for quasar radiation in a line encounters insuperable difficulties involving the energy of the quasar and the sizes of the radiating regions.Translated fromAstrofizika, Vol. 37, No. 3, 1994.     

A global solution in the resonance problem of Poincaré

Poincaré's procedure for the construction of a global solution for a particular class of resonance problem is investigated, with particular emphasis placed on those motions corresponding to circulation in the phase space. It is demonstrated that an error on Poincaré's part leads to an impractical, yet formally acceptable, procedure.The merits of alternative methods are discussed, with particular reference to the studies of Garfinkel and the author.Presented at the Conference on Celestial Mechanics, Oberwolfach, Germany, August 27–September 2, 1972.     

Transfers between libration-point orbits in the elliptic restricted problem

A strategy is formulated to design optimal time-fixed impulsive transfers between three-dimensional libration-point orbits in the vicinity of the interiorL ₁ libration point of the Sun-Earth/Moon barycenter system. The adjoint equation in terms of rotating coordinates in the elliptic restricted three-body problem is shown to be of a distinctly different form from that obtained in the analysis of trajectories in the two-body problem. Also, the necessary conditions for a time-fixed two-impulse transfer to be optimal are stated in terms of the primer vector. Primer vector theory is then extended to non-optimal impulsive trajectories in order to establish a criterion whereby the addition of an interior impulse reduces total fuel expenditure. The necessary conditions for the local optimality of a transfer containing additional impulses are satisfied by requiring continuity of the Hamiltonian and the derivative of the primer vector at all interior impulses. Determination of the location, orientation, and magnitude of each additional impulse is accomplished by the unconstrained minimization of the cost function using a multivariable search method. Results indicate that substantial savings in fuel can be achieved by the addition of interior impulsive maneuvers on transfers between libration-point orbits.An earlier version was presented as Paper AAS 92–126 at the AAS/AIAA Spaceflight Mechanics Meeting, Colorado Springs, Colorado, February 24–26, 1992.     

A simple problem of radiative transfer by multilevel atoms

We consider the problem of determining the radiation fields reflected and transmitted by a slab containing multilevel hydrogen atoms and illuminated on one side by a given radiation field. We treat the extreme non-LTE situation in which the populations of the different levels are determined by the radiative processes. We take into account the population and the transfer effects in a self-consistent way by solving the transfer equations in all the lines and continua together with the equations of statistical equilibrium for all levels. We limit ourselves to the idealistic case of rectangular profiles in the lines and continua and to a model of atoms with 4 levels and a continuum. Under conditions close to thermodynamic equilibrium we empirically derive a Schuster-like law for the continua with transmitted radiation fields varying as the inverse of the optical thickness. Turning to out-of-equilibrium conditions we emphasize the crucial role of the loss probability of the Ly photons. Owing to the rapid decrease of the excitation/ionization degree in the medium and contrary to the conservative case the optical thicknesses of the subordinate transitions now remain finite even when the population of the fundamental level along the line-of-sight becomes infinite. As a result of this relative transparency the strong emission lines formed by recombination mechanisms can escape from the medium. Although the present problem remains largely academic because of the number of simplifications introduced we suggest some possible applications and developments.Published in Astrofizika, Vol. 37, No. 2, pp. 313–338, April–June, 1994.     

A problem in resonance

Saturn's satellite Hyperion experiences large perturbations by Titan, the largest of Saturn's satellites, because of the closeness of Hyperion's orbital period to three-quarters of that of Titan.The motion of Hyperion is a superposition of periodic fluctuations (both free and forced) onto a motion which is periodic in a suitable uniformly rotating frame of reference, and in which Hyperion would be at a maximum distance from Saturn at each conjunction with Titan.Successive attempts to determine the mass of Titan from observed perturbations of Hyperion have suffered from omissions in the theory of terms subsequently found to be significant.An attempt is in progress which, it is believed, comprises all long-period changes in the osculating elements which are of second degree in the mass of Titan, and of third degree in the eccentricity of its orbit. Results so far obtained indicate that the period of the free motion of the orbit plane of Hyperion is better determined by Woltjer's reduction of the observations than by his theoretical calculations.Presented at the Conference on Celestial Mechanics, Oberwolfach, Germany, August 27–September 2, 1972.     

Rotation of pre-main-sequence stars in NGC 2264 and the solar angular momentum problem

Preliminary measurements of rotational velocities of pre-main sequence stars indicate that stars evolving into early F or late A spectral type have rotational velocities which are consistent with present Main-Sequence stars of similar spectral type. Stars evolving into G type, however, have rotational velocities which are as high as 100 km s^–1 and would reach the Main Sequence with velocities of 150 km s^–1. This requires the presence of a strong stellar wind to carry off considerable angular momentum in order to slow down the Sun to its present low rotational velocity.Paper presented at the Conference on Protostars and Planets, held at the Planetary Science Institute, University of Arizona, Tucson, Arizona, between January 3 and 7, 1978.     

Discrepant measurements of the radial velocities of SS cygni: solution to the problem

We show that the discrepancies among the various measurements of the radial velocity curve of the K5 V star in the dwarf nova SS Cyg have been caused by (1) poor choices of lines for measuring its velocity and (2) large distortions in its velocity caused by heating from the white dwarf and its accretion disk. The correctK velocity of the center of mass of the K5 V star is 158±3km s ^–1.Paper presented at the IAU Colloquium No. 93 on Cataclysmic Variables. Recent Multi-Frequency Observations and Theoretical Development, held at Dr. Remeis-Sternwarte Bamberg, F.R.G., 16–19 June, 1986.     

The UV continuum 1450–2100 Å and the problem of the solar temperature minimum

This study is based on a set of ten solar rocket spectra well exposed for photometry photographed on July 27, 1966 by Purcell, Snider, and Tousey.The photometry of the far UV continuum illustrates the transition of the solar temperature minimum at 1700 Å in the solar spectrum - (a) the continuum intensity decreases by 30–50% between 1700 Å and the¹ D limit of silicon at 1682 Å, and (b) the equivalent brightness temperature shows minimum values throughout the spectral range 1540–1682 Å, which average just under 4700 ± 100K.The minimum UV brightness temperature is 300K higher than the far infrared measurement of the solar minimum temperature, and possible reasons for this are discussed.Brightness temperatures measured in prominent CO band heads and in the aluminum 1937 Å auto-ionization line also are given.     

A new regularization of the planar problem of three bodies

A new method of simultaneously regularizing the three types of binary collisions in the planar problem of three bodies is developed: The coordinates are transformed by means of certain fourth degree polynomials, and a new independent variable is introduced, too. The proposed transformation is in each binary collision locally equivalent to Levi-Civita's transformation, whereas the singularity corresponding to a triple collision is mapped into infinity. The transformed Hamiltonian is a polynomial of degree 12 in the regularized variables.Presented before the Division of Dynamical Astronomy at the 133rd meeting of the American Astronomical Society, Tampa, Florida, December 6–9, 1970.Department of Aerospace Engineering and Engineering Mechanics.