Optically thick accretion onto black holes

Spherically symmetric, steady-state, optically thick accretion onto a nonrotating black hole with the mass of is studied. The gas accreting onto the black hole is assumed to be a fully ionized hydrogen plasma withn ₀=10⁸ cm^–3 andT ₀=10⁴ K far from the black hole, and a new approximate expression for the Eddington factor is introduced. The luminosity is estimated to beL=1.875×10³³ erg s^–1, which primarily arises from the optical surface (1) ofT10⁴ K. The accretion flow is characterized by 1 and (v/c)10. In the optically thin region, the flow remains isothermal, and the increase of temperature occurs at 1. The radiative equilibrium is strictly realized at (v/c)10.     

The asymptotic behavior of the supersonic solutions of the two-fluid solar wind equations

Three different asymptotic branches of the two-fluid equations are found with _e ^m , _p ⁿ , where, is the inverse distance from the Sun, with (m, n) = (2/7, 2/7), (2/7, 6/7), (4/3, 4/3); other special solutions are also found but they correspond to special choices of density and temperature at the corona. In all the (4/3, 4/3) solutions, the electron and proton temperatures tend to equality at large distances.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

Planetary accretions in jet streams

The problem of planetary accretion in a jet stream is studied using the model developed by Alfvén and Arrhenius. We find that there are basically three types of planetary accretion: namely, fast process _c < _i , slow process _c ~ _i and delayed process _c > _i where _c is the characteristic time of the occurrence of catastrophic accretion and _i the time-scale of mass injection to the planetary system (3×10⁸ yr). These different time scales of accretion are found to be closely related to the primordial thermal profiles and equatorial inclinations of the planets. Finally, Venus' retrograde rotational spin is shown to be a possible result of accretion process in a jet stream.     

Dimensional Analysis of the Flare Distribution Problem

Dimensional analysis is used to derive the distribution of solar flare energies,p() = A-^3/2, in accordance with recent observational and numerical results. Several other scalings, notably _fl ² , where _fl is the flare duration, are obtained as well.     

Gravitational radiation and spiralling time of close binary systems (V)

Forty-six binary systems with their primary component masses between 2M and 3M have been considered for gravitational radiation study. Power output by gravitational radiation (P _B ) and spiral time ₀ for all individual systems have been evaluated. A relation has been given betweenP _B and ₀. The rate of decrease of orbital period (P) has also been given for 10 eccentric orbit systems.     

On the solar oblateness: The combined effect of a pole-equator difference in effective temperature and mechanical heating

With the help of a model atmosphere of the Sun we evaluate the pole-equator difference in flux (as measured by Dicke and Goldenberg) assuming the following type of pole-equator temperature difference (T=T _e –T _p ): (a) T 2K for > ₀ (₀ 0.05); (b) T 10K for < ₀.The small T at all optical depths given by (a) could, for example, be due to a pole-equator difference in effective temperatures. At small optical depths a difference in mechanical heating could give rise to the larger temperature difference given by (b). We compare the results of our calculations with Dicke and Goldenberg's observations.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

A systematic method for the analysis of high-resolution fraunhofer line profiles

A fraunhofer line profile depends on various parameters, partly related to the photospheric structure (T, P _g, P _e, v _conv, v _turb), partly to the atom or ion involved (such as oscillator strength, energy levels), partly also resulting from the interaction of the relevant kind of particles with the photosphere, and the photospheric radiation field. In this paper we shall mainly pay attention to the determination of: the macroturbulent (convective) velocities, v _conv (); the damping constant (); the abundance, A _el; the distribution function (v _conv, ) of the convective velocities at each depth ; the source function, S (); the microturbulent velocities, v _turb ().The particular difficulty with these unknowns is that they are, as a rule, coupled in the resulting line profiles, that is: the shapes and intensities in these profiles are determined by the combined influence of these unknowns (together with the other above-given parameters).In this paper we describe a method to determine these six unknowns empirically by separating them, in analysing accurate high-resolution observations of line profiles of a multiplet. The unknown functions and quantities are consecutively determined in the above given succession. For each determination another, appropriate part of the line profile is used. In some cases the influence of the mutual coupling of the various parameters cannot be completely eliminated, and an iterative method has to be used.The method is summarized in Table II and section 2, and is further explained in sections 3 to 8. It is applied to an infrared Ci multiplet. The main results are the following:     

High order symplectic integrators for perturbed Hamiltonian systems

A family of symplectic integrators adapted for the integration of perturbed Hamiltonian systems of the form H=A+B was given in (McLachlan, 1995). We give here a constructive proof that for all integer p, such integrator exists, with only positive steps, and with a remainder of order O(^p + ²²), where is the stepsize of the integrator. Moreover, we compute the analytical expressions of the leading terms of the remainders at all orders. We show also that for a large class of systems, a corrector step can be performed such that the remainder becomes O(^p +⁴²). The performances of these integrators are compared for the simple pendulum and the planetary three-body problem of Sun–Jupiter–Saturn.     

Gravitational radiation and spiralling time of close binary systems (IV)

Binary systems with their primary and secondary component masses less than 2M have been investigated to evaluate the rate of emission of gravitational energy (P _B) and spiralling time (₀) for them. In all twenty-two binary systems have been considered. It is found that in spite of the same mass range, these systems form two distinct groups. New relations have been given betweenP _B and (₀) for each group. For a few eccentric orbit systems the rate of decay of orbital periods due to the loss of energy from the system via gravitational radiation emission has also been given and compared with a short-period binary pulsar.     

On Geovenelli's approximate form of j(τ)

The accuracy of the approximate replacement of the frequency meanj() of the total intensityJ () byJ _c+a{J ₀()–J _c} by Geovenelli, whereJ _c is the total intensity at the continuum,J ₀() is that at the line center anda is independent of depth, is studied. Numerical calculations for strong solar lines show very high values of the residual intensities. A slight change in the approximation is then suggested to get a result quite consistent with observational and existing calculated data.     

A simplified model solar atmosphere

A simplified representation of the temperature distribution in the solar photosphere is proposed: ( ₀) = ₀ - ₁ log ₀. An expression is derived for the emergent continuous spectrum from the simple model. The limitations and applications of the simple model are discussed.     

On the A-transform of the exponential integrals

In this paper, a technique of recursive analysis is developed for the integral transform A of the exponential integral functionsE _n which is denoted as _n (). The main result of this analysis enables us to establish a two-term recurrence formula for _n (0) and a three-term recurrence formula for _n (); 0. A computational algorithm based on these formulae is also constructed and its numerical results forn=2(1)25 are presented to 15-digit accuracy.     

Physical Properties of Dust in the Martian Atmosphere: Analysis of Contradictions and Possible Ways of Their Resolution

Atmospheric aerosols play an important role in forming the Martian climate. However, the basic physical properties of the Martian aerosols are still poorly known; there are many contradictions in their estimates. We present an analytical overview of the published results and potentialities of various methods. We consider mineral dust. Zonally averaged data obtained from mapping IR instruments (TES and IRTM) give the optical thickness of mineral aerosols ₉ = 0.05–0.1 in the 9-m band for quite atmospheric conditions. There is a problem of comparing these estimates with those obtained in the visible spectral range. We suggest that the commonly used ratio _vis/₉ >2 depends on the interpretation and it may actually be smaller. The ratio _vis/₉ 1 is in better agreement with the IRIS data (materials like montmorillonite). If we assume that _vis/₉ = 1 and take into account the nonspherical particle shape, then the interpretation of ground-based integrated polarimetric observations ( < 0.04) can be reconciled with IR measurements from the orbit. However, for thin layers, the sensitivity of both methods to the optical thickness is poorly understood: on the one hand, polarimetry depends on the cloud cover and, on the other hand, the interpretation of IR measurements requires that the atmospheric temperature profile and the surface temperature and emissivity be precisely known. For quite atmospheric conditions, the local optical-thickness estimates obtained by the Bouguer–Lambert–Beer method and from the sky brightness measured from Viking 1 and 2 and Mars Pathfinder landers are much larger: = 0.3–0.6. Estimates of the contrasts in images from theVikingorbiters yield the same values. Thus, there is still a factor of 3 to 10 difference between different groups of optical-thickness estimates for the quiet atmosphere. This difference is probably explained by the contribution of condensation clouds and/or by local/time variations.     

General integral transform of the exponential integrals

General integral transform of the exponential integralsE _n is considered and will be denoted asB _(k) ⁿ (). Different expressions and the equations satisfied byB _(k) ⁿ are developed. Two-term recurrence formula forB _(k) ⁿ (0) and three-term recurrence formula forB _(k) ⁿ (); 0 will be established for a givenk1 andn=2,3, ...,N. The computational algorithms based on these formulae are also constructed for the casesk=1,2,3, andn2. Finally the numerical results fork=2,3 andn=2(1)25 are presented to 15-digit accuracy     

Tidal circularization in massive binaries

We consider the time-scale of orbital circularization for those massive binary systems that produce WR+C systems. Using the observation that such systems have acquired circular orbits on a time-scale of the order of the Main-Sequence lifetime of the initial secondary, we estimate the intrinsic time-scale of the tidal interaction ₀. We find that the tidal dissipation in such systems is very efficient and that, on the average, ₀ ~ 10³ yr.     

A novel methodology for radiative transfer in a planetary atmosphere

A novel methodology for evaluating the field of anisotropically scattered radiation within a homogeneous slab atmosphere of arbitrary optical thickness is provided. It departs from the traditional radiative transfer approach in first considering that the atmosphere is illuminated by an isotropic light source. From the solution of this problem, it subsequently proceeds to that for the more conventional case of monodirectional illumination. The azimuthal dependence of the field is separated in the usual manner by an harmonic expansion, leaving a problem in four dimensions (=optical depth, ₀=thickness, , =directions of incidence and scattering) which, as is well known, is numerically extremely inconvenient. Two auxiliary radiative transfer formulations of increasing dimensionality are considered: (i) a transfer equation for the newly introduced functionb ^m(,,₀) with Sobolev's function ^m(,₀) playing the role of a source-function. Because the incident direction does not intervene, ^m is simply expressed as a single integral term involvingb ^m. For bottom illumination, an analogous equation holds for the other new functionh ^m(,,₀). However, simple reciprocity relations link the two functions so that it is only necessary to considerb ^m; (ii) a transfer equation for the other new functiona ^m(,,,₀) with a source-function provided by Sobolev's functionD ^m(,,₀). For bottom illumination, another functionf ^m(,,,₀) is introduced; by a similar argument using reciprocity relations,f ^m is reduced toa ^m rendering necessary only the consideration ofa ^m. However, a fundamental decomposition formula is obtained which shows thata ^m is expressible algebraically in terms of functions of a single angular variable. The functions ^m andD ^m are shown to be the values in the horizontal plane ofb ^m anda ^m, respectively. The other auxiliary functionsX ^m andY ^m are also expressed algebraically in terms ofb ^m. These results enable one to proceed to the final step of evaluating the radiation field for monodirectional illumination. The above reductions toalgebraic relations involving only the functionb ^m appear to be more advantageous than Sobolev's (1972) recent approach; they also circumvent some basic numerical difficulties in it. We believe the present approach may likewise prove to be superior to most (if not all) other methods of solution known heretofore.This paper presents the results of one phase of research carried out at the Jet Propulsion Laboratory under Contract No. NAS-7-100 sponsored by the National Aeronautics and Space Administration.     

Analysis of extreme-ultraviolet spectroheliograms of solar prominences

The optical depth at the head of the Lyman continuum, _H, is determined at a number of positions in three hedgerow prominences using spectroheliograms (5 × 5 resolution) of C III 977, LC 896, and O IV 554 observed with the Harvard experiment on Skylab. At heights greater than 10 above the limb the maximum value of _H is 30 to 50, which occurs at the central part of the prominences. For one of the prominences the determination of _H is found to be consistent with data from spectroheliograms of Mg X 625. The degree of ionization of hydrogen is estimated from the intensity of LC 896 at _H 1. In the central part of a model prominence N _P/N _HI1.9 for a reasonable range of the electron densities, where N _P and N _HI are the proton density and the neutral hydrogen density, respectively.     

Transformations to extend the range of application of power series solutions of differential equations of motion

Let the solution of a differential equation, expanded in powers of the independent variablet, have radius of convergenceT. let , wheret=t(), be a new independent variable, and let the corresponding power series in have radius of convergenceS. Thent(S) will not in general be equal toT. Ift(S)>T, then the series in powers of may have advantages over those in powers oft. Mathematical consequences of this distinction have been appreciated since the time of Poincaré. In this note the practical applications of some transformations are investigated.     

Structure and physics of solar faculae

Taking into account the effect of roughness (or local departures from sphericity) of the emitting layers in the chromosphere-corona transition zone (CCT) allows one to determine the optical depths of layers responsible for resolved structures in Cii, Ciii, Oiv, and Ovi lines. The result, at the top of the irregularities, is of the order of respectively ₁ 3.5,2.0, 1.6,0.5, and for the bottom of these irregularities, ₂ = 0.7, 0.4, 0.3, 0.25. The characteristic angle of these irregularities is, respectively, of the order of 35 °, 33 °, 35 °, and 41 °. For unresolved structures of Civ and Ovi (already analyzed in the spherical symmetry hypothesis in Paper III), one finds ₁ 0.6; 0.9 and ₂ 0.12; 0.2 in the case of quiet areas; in the case of active areas, the range is broader for Civ and Ovi, from 1.0 to 1.7 for ₁ and from 0.2 to 0.9 for ₂. The values obtained from Ovi are in reasonable agreement with each other for resolved and unresolved structures. And the obtained values of ₁ and ₂ correspond not too badly with the determinations made in Paper III, by methods not exceedingly influenced by the spherical symmetry hypothesis.     

Effects of diverging coronal fields on the solar wind expansion

The expansion of the solar wind in divergent flux tubes is calculated by taking into account a magnetic acceleration of the particles, analogous to the magnetic mirror effect.The resulting force term included in the magnetohydrodynamical equations describes a conversion of thermal into kinetic energy. This causes an additional acceleration of the solar wind plasma which has never been taken into account before. The force is directed opposite to the magnetic field gradient. Consequently, in this case the solar wind velocity increases faster to its asymptotic value than it does for corresponding nonmagnetic solutions. Therefore inside and close to the solar corona markedly higher velocities are found. Compared to strictly hydrodynamical models, the critical point is shifted towards the Sun, and the radial decrease of the ratio of thermal to kinetic energy is faster.The necessary prerequisites for these calculations are (a) that the gyroperoid _g of the plasma particles is much shorter than the Coulomb collision time _c , and (b) that the collision time _c is shorter than the characteristic time _d in which an appreciable amount of thermal anisotropy is built up. Thus it is (a) insured that the particles have established magnetic moments and follow the guiding center approximation, and (b) an almost isotropic velocity distribution function is maintained which, in this first approximation of a purely radial expansion, justifies the use of isotropic pressures and temperatures.Both (a) and (b) are shown to be fulfilled in a region around the Sun out to about 20R , and thermal anisotropies developing outside of this region could explain the observed magnetically aligned anisotropies at 1 AU.