Evolutionary processes in protoplanetary accretion disks: the propagation of axisymmetric shock waves

In this paper we investigate both the global and the local hydrodynamics of axisymmetric accretion disks around young stellar objects under the simultaneous action of viscosity, self-gravity and pressure forces. For simplicity, we take for the global model a polytropic equation of state, make the infinitely thin disk approximation and characterize the surface density and temperature profiles in the disk as power laws in the radial distance r from the protostar. We solve the problem of the general density profile of a Keplerian disk showing that self-gravity could not be an important factor for the fast formation of the rocky cores of giant gaseous planets in our solar system. Under the hypothesis that the unperturbed rotation curve of the disk is nearly Keplerian throughout the radial extent, we can estimate with our polytropic model a lower limit for the resulting masses M_d(r) of stable disks up to 100 AU. These masses are in the range of the so-called minimum mass solar nebular (_d/M_s ≈ 0.01–0.02).By adopting a simplified viscosity model, where the height-integrated turbulent dynamical viscosity ν is a function of the surface density σ like η ∝ σ^Γ, we derive in the local shearing sheet model linearized evolution equations for small density perturbations describing both a diffusion process and the propagation of acoustic density waves. We solve a special initial value problem and calculate the appropriate Green's function. The analytical solutions so obtained describe in the case Γ < 0 the successive formation of quasi-stationary ring-shaped density structures in a disk with a definite mode of maximum instability, whereas in the case Γ > Γ_c the density wave equation describes the propagation of an “overstable” ring-shaped acoustic density wavelet to the outer ranges of the accretion disk. Whereas the group velocity of the wave packet is subsonic, the phase velocities of individual wave crests in the wave packet are supersonic. The mode of maximum instability, the growth rate and the number of growing waves in the wavelet are controlled by Γ and α. Our present knowledge concerning turbulent viscosity in protoplanetary disks is not sufficient to decide whether or not the case Γ > Γ_c is realized.The suggested structuring processes in the linear theory should initiate in the non-linear regime the formation of narrow ring-shaped density shock waves moving through the protoplanetary disk. These non-linear waves could produce extremely spatially and temporally heterogeneous temperature regions in the disk. We speculate that ring-shaped density waves, excited by inner boundary conditions and which have dominated the disk's evolution at early times, are responsible both for the fast growth of dust to planetesimals and at least for the rapid accretion of the rocky cores of giant gaseous planets in the protoplanetary accretion disk (shock wave trigger hypothesis). We derive provisional scaling rules for planetary systems regarding the spacing of orbits as a function of the mass ratio of the protoplanetary disk to the protostar. However, further analytical work and linear as well as nonlinear numerical simulations of density waves excited by inner boundary conditions are needed to consolidate the results and speculations of our linear wave mechanics in the future.     

Exoplanet's Figure and Its Interior

Along with the development of the observing technology, the observation and study on the exoplanets’ oblateness and apsidal precession have achieved significant progress. The oblateness of an exoplanet is determined by its interior density profile and rotation period. Between its Love number k₂ and core size exists obviously a negative correlation. So oblateness and k₂ can well constrain its interior structure. Starting from the Lane-Emden equation, the planet models based on different polytropic indices are built. Then the flattening factors are obtained by solving the Wavre's integro-differential equation. The result shows that the smaller the polytropic index, the faster the rotation, and the larger the oblateness. We have selected 469 exoplanets, which have simultaneously the observed or estimated values of radius, mass, and orbit period from the NASA (National Aeronautics and Space Administration) Exoplanet Archive, and calculated their flattening factors under the two assumptions: tidal locking and fixed rotation period of 10.55 hours. The result shows that the flattening factors are too small to be detected under the tidal locking assumption, and that 28% of exoplanets have the flattening factors larger than 0.1 under the fixed rotation period of 10.55 hours. The Love numbers under the different polytropic models are solved by the Zharkov's approach, and the relation between k₂ and core size is discussed.     

Effects of rotation and tidal distortions on the shapes of radial velocity curves of polytropic models of pulsating variable stars

Anharmonic oscillations of rotating stars have been studied by various authors in literature to explain the observed features of certain variable stars. However, there is no study available in literature that has discussed the combined effect of rotation and tidal distortions on the anharmonic oscillations of stars. In this paper, we have created a model to determine the effect of rotation and tidal distortions on the anharmonic radial oscillations associated with various polytropic models of pulsating variable stars. For this study we have used the theory of Rosseland to obtain the anharmonic pulsation equation for rotationally and tidally distorted polytropic models of pulsating variable stars. The main objective of this study is to investigate the effect of rotation and tidal distortions on the shapes of the radial velocity curves for rotationally and tidally distorted polytropic models of pulsating variable stars. The results of the present study show that the rotational effects cause more deviations in the shapes of radial velocity curves of pulsating variable stars as compared to tidal effects.     

Comments on ‘the origin of the Earth—Moon system’

The main points are presented of a new hypothesis of the origin of the Earth—Moon system, developed on the basis of Savi's (1961) theory of the origin of rotation of celestial bodies. The cooling off and contraction due to gravitational attraction on vast particle systems, with the pushing out of electrons from atom shells result in a continually increasing density. Depending on the amount of mass, this pushing out can lead to the expulsion of electrons and the creation of a magnetic field by which a rotational motion is brought about. These conditions are satisfied for the Earth's mass and all larger masses. If the Earth and the Moon formed a unique body, the protoplanet, then once rotational motion had begun, the primeval spherical body must have taken the shape of a large Jacobi ellipsoid. New condensation followed, however no longer solely around the centre of the protoplanet, but also along the edge of the ellipsoid, the process leading to the creation of the dual Earth—Moon system.     

The galactic centre (invited review)

Recent observations of the galactic centre at radio and infrared wavelengths are reviewed. Particular emphasis is placed on the new large scale radio structures — the galactic centre lobe, and the filamentary ribbon structures in the radio arc — which have just been reported. New observations of the two parsec ring of molecular hydrogen, and the relationship of the ring to the interior distribution of ionised gas are discussed. The kinematis and distribution of neutral material, shocked molecular hydrogen and plasma in the inner few parsecs of the nucleus are considered in terms of a central luminous mass-losing object, together with rotation and infall of the neutral material.Paper presented at the IAU Third Asian-Pacific Regional Meeting, held in Kyoto, Japan, between 30 September–6 October, 1984.     

Expression of the initial Poincaré canonical variables as functions of the new in a ninth order J-S theory

We establish the solution of the ninth order — in masses — canonical J-S equations of motion by Hori-Lie technique — i.e., by expressing the initial Poincaré canonical variables as functions of the new variables through the Hori-Lie canonical transformation. Terms of order higher than 9 in the masses are neglected.     

Similarity between the structure and stability of isothermal and polytropic gas spheres

As shown by Chiéze, polytropic gas spheres imbedded in an ambient medium with a given pressure exhibit a relation between the mass and radius similar to isothermal gas spheres, providedn(polytropic index)>3 or <–1. Here, further investigation is made of the polytropic spheres. Not only the relation between radius and external pressure, but the relation between the central density ( _c ) and the mass (M) is similar to isothermal spheres and neutron star models. Dynamical calculation shows that the stability of polytropic spheres is lost at the configuration which corresponds to the first peak in the _c —M relation, as is the case for neutron star models in general relativity.     

High-Precision Ephemerides of Planets—EPM and Determination of Some Astronomical Constants

The latest version of the planetary part of the numerical ephemerides EPM (Ephemerides of Planets and the Moon) developed at the Institute of Applied Astronomy of the Russian Academy of Sciences is presented. The ephemerides of planets and the Moon were constructed by numerical integration in the post-Newtonian metric over a 140-year interval (from 1880 to 2020). The dynamical model of EPM2004 ephemerides includes the mutual perturbations from major planets and the Moon computed in terms of General Relativity with allowance for effects due to lunar physical libration, perturbations from 301 big asteroids, and dynamic perturbations due to the solar oblateness and the massive asteroid ring with uniform mass distribution in the plane of the ecliptic. The EPM2004 ephemerides resulted from a least-squares adjustment to more than 317000 position observations (1913–2003) of various types, including radiometric measurements of planets and spacecraft, CCD astrometric observations of the outer planets and their satellites, and meridian and photographic observations. The high-precision ephemerides constructed made it possible to determine, from modern radiometric measurements, a wide range of astrometric constants, including the astronomical unit AU = (149597870.6960 ± 0.0001) km, parameters of the rotation of Mars, the masses of the biggest asteroids, the solar quadrupole moment J ₂ = (1.9 ± 0.3) × 10⁻⁷, and the parameters of the PPN formalism β and γ. Also given is a brief summary of the available state-of-the-art ephemerides with the same precision: various versions of EPM and DE ephemerides from the Jet Propulsion Laboratory (JPL) (USA) and the recent versions of these ephemerides—EPM2004 and DE410—are compared. EPM2004 ephemerides are available via FTP at ftp://qua-sar.ipa.nw.ru/incoming/EPM2004.__________Translated from Astronomicheskii Vestnik, Vol. 39, No. 3, 2005, pp. 202–213.Original Russian Text Copyright © 2005 by Pitjeva.     

The Roche Coordinates in three dimensions and their application to problems of double-star astronomy

In a preceding paper (Kopal, 1969; in what follows referred to as Paper I) we introduced a new system of curvilinear coordinates-hereafter referred to as Roche Coordinates — in which spheres of constant radius in spherical polars have been replaced by surfaces of constant potential of a rotating gravitational dipole; while the angular coordinates are orthogonal to the equipotentials. In Paper I we established an explicit form of such a transformation, and related the Roche coordinates with polar coordinates (with which they coalesce in the immediate neighbourhood of each one of the two finite mass-points) in the plane case. The aim of the present investigation will be to generalize the definition of the Roche coordinates to three dimensions.The opening Section 1 of this paper will contain a general outline of the proposed three-dimensional transformation; and in Section 2 details of this transformation will be explicitly worked out correctly to quantities of first order in superficial distortion — an approximation which should prove adequate in regions surrounding the two finite masses; while in Section 3 we shall evaluate (to this degree of accuracy) the metric coefficients of the respective transformation, and its direction cosines, in both polar and curvilinear coordinates. Section 4 will then contain a formulation of the fundamental equations of hydrodynamics in terms of the three-dimensional Roche coordinates; and their advantages for a treatment of certain classes of dynamical problems encountered in doublestar astronomy will be illustrated in the concluding Section 5 by an investigation of the vibrational stability of the Roche model. We shall show that this model is capable of performing free radial oscillations which remain barotropic only if its equilibrium form is spherical (i.e., in the absence of any external mass in the neighbourhood); but not if it is distorted to any extent by rotation or tides.     

Gravitational Radiation of Relativistic <Emphasis Type="Italic">n</Emphasis> = 1 Polytropes

The gravitational radiation of n = 1 polytropes undergoing quasiradial pulsations is examined. The intensity of the gravitational radiation and the gravitational wave amplitudes are calculated for polytropic models of white dwarfs and neutron stars when the energy of rotation of the object serves as the source of the radiated energy. Calculations of h₀ show that objects with a polytropic equation of state can describe the expected gravitational radiation from white dwarfs and neutron stars. The gravitational radiation of polytropic models of galactic nuclei and quasars is also examined. These objects can create a high enough background of gravitational radiation at frequencies of 10^-8–10^-11 Hz for gravitational wave detectors operating in this frequency range. __________ Translated from Astrofizika, Vol. 48, No. 4, pp. 603–612 (November 2005).     

The design of the Central-Axis Reflector (ZAS)

The Central-Axis Reflector, the design principle which is presented below is a segmented-mirror telescope. The inventions relate mainly to the optical system and to the tracking apparatus.A large number of small individual mirror bodies, ground off-axis (hexagonal/polygonal) produce one primary mirror with closed circular aperture when joined together.The overall design of the tracking apperatus results directly —and thus without unnecessary adornment — from the two planes of to aminimum but which are required for tracking of the telescope.Paper presented at the Symposium on the JNLT and Related Engineering Developments, Tokyo, November 29–December 2, 1988.     

Companions' influence on the emission-line rotation curves of galaxies

The sample of members of isolated pairs of galaxies and close binary interacting systems — demonstrating a systematic falling of the emission-line rotation curves after the maxima — are considered. It is found that there is a significant correlation between the characteristics of the galaxies' emission-line velocity fields and the spatial distance to the companion. The different factors affecting the investigated relationships are discussed.     

Some Problems with Slow Rotation of CP Stars

Some difficulties in explaining the slow rotation of CP stars are discussed. The most likely hypotheses are (1) a loss of angular momentum involving a magnetic field during “pre-main sequence” evolution and (2) the slow rotation existed from the very start of the creation of these stars. The braking hypothesis is supported by only one property of CP stars— the lower the mass of the star is, the greater the difference between its average rotation velocity vsini and that of normal stars. On the other hand, there is another property— the lower the rotation speeds of CP stars are, the greater their fraction among normal stars. The latter property supports the hypothesis that the lower the initial rotation speed of a star is when it is created, the greater the probability will become chemically peculiar. This property is inherent in chemically peculiar stars both with and without a magnetic field. It is proposed that the cause of the slow rotation of CP stars must be sought in the very earliest phases of their formation, as should the cause of the separation into chemically peculiar magnetic, chemically peculiar nonmagnetic, and normal stars.__________Translated from Astrofizika, Vol. 48, No. 2, pp. 229–245 (May 2005).     

The search for brown dwarfs with infrared surveys

I compute the maximum number of observable brown dwarfs for various infrared surveys by combining the maximum possible Oort limit (0.1 missing Mpc^–3 with all possible brown dwarf mass and age distributions. This approach shows what limits will be placed on the contribution of brown dwarfs to any possible missing mass if no brown dwarfs are observed. I consider brown dwarfs with masses of 0.01—0.08 M and ages of 10⁹—10¹⁰ years.The full range of predicted numbers of brown dwarfs above 6 times the noise of each of the below surveys is: IRAS Point Source Catalog 0.02—6 IRAS Faint Source Catalog |b| > 10° 0.05—16 ISO (2 week 12µm survey) 0.15—80 SIRTF (2 week 12µm survey) 2.50—1600 WIRE (4 month 12µm survey) 21.80—6000 DENIS (half sky) |b| > 10° 0.00—2000 2MASS (full sky) |b| > 10° 0.00—8000A failure to find brown dwarfs in the IRAS FSC would just barely rule out about half of the mass—age range for Oort limit total masses. A failure to find brown dwarfs in 2MASS/DENIS would rule out roughly the same mass—age range, but would set a limit of 0.1—0.01 times the Oort mass in that mass—age region. No limits would be set for the other half of the mass—age range since both IRAS and 2MASS/DENIS have insufficient sensitivity for brown dwarfs with T < 750 K.A failure to find brown dwarfs with ISO would rule out almost all of the mass—age range for Oort limit total masses, but would not set a significantly lower limit to the brown dwarf mass limit. A failure to find brown dwarfs with SIRTF or WIRE would rule out the entire mass—age range for Oort limit total masses and set an upper limit of 0.1—0.001 times the Oort mass.To date, about 18% of the IRAS FSC has been searched down to 6, and no brown dwarfs have been found. This sets a 95% upper limit of 3 in 18% of the sky, or 13 in the entire FSC for |b| > 10°. To begin to set useful limits from 2MASS or DENIS, approximately 400 square degrees needs to be analyzed. To date, only a few square degrees of results from the 2MASS prototype camera have been examined, with no brown dwarfs found so far.     

Non-uniformly rotating,self-gravitating,compressible masses with internal meridian circulation

The structure of self-gravitating, inviscid, compressible fluids is investigated assuming a polytropic relation between pressure and density. A class of solutions with non-uniform rotation and internal meridian circulation are presented and the stream lines of the flow calculated using a perturbation technique.     

Tidal evolution in close binary systems

The aim of the present paper will be to give a mathematical outline of the theory of tidal evolution in close binary systems of secularly constant total momentum — an evolution activated by viscous friction of dynamical tides raised by the two components on each other. The first section contains a general outline of the problem; and in Section 2 we shall establish the basic expressions for the energy and momenta of close binaries consisting of components of arbitrary internal structure. In Section 3 we shall investigate the maximum and minimum values of the energy (kinetic and potential) which such systems can attain for given amount of total momentum; while in Section 4 we shall compare these results with the actual facts encountered in binaries with components whose internal structure (and, therefore, rotational momenta) are known to us from evidence furnished by the observed rates of apsidal advance.The results show that all such systems — be these of detached or semi-detached type — disclose that more than 99% of their total momenta are stored in the orbital momentum. The sum of the rotational momenta of the constituent components amounts to less than a percent of the total — a situation characteristic of a state close to the minimum energy for given total momentum. This appears, moreover, to be true not only of the systems with both components on the Main Sequence, but also of those possessing evolved components in contact with their Roche limits.Under such conditions, a synchronism between rotation and revolution (characteristic of both extreme states of maximum and minimum energy) is not only possible, but appears to have been actually approached — if not attained — in the majority of cases. In other words, it would appear that — in at least a large majority of known cases — the existing close binaries have already attained orbits of maximum distension consistent with their momenta; and tidal evolution alone can no longer increase the present separations of the components to any appreciable extent.The virtual absence, in the sky, of binary systems intermediate between the stages of maximum and minimum energy for given momentum leads us to conjecture that the process of dynamical evolution activated by viscous tides may enroll on a time-scale which is relatively short in comparison with their total age — even for systems like Y Cygni or AG Persei, whose total age can scarcely exceed 10⁷ yr. A secular increase of the semi-major axes of relative orbits is dynamically coupled with a corresponding variation in the velocity of axial rotation of both components through the tidal lag arising from the viscosity of stellar material. The differential equations of so coupled a system are given in Section 5; but their solution still constitutes a task for the future.The Lunar Science Institute Contribution No. 90. The Lunar Science Institute is operated by the Universities Space Research Association under Contract No. NSR 09-051-001 with the National Aeronautics and Space Administration.     

太阳多方模型对行星轨道要素变化的长期摄动影响

本文利用天体力学中的摄动理论和天体物理学中的气体星多方模型理论研究了太阳多方模型对行星轨道要素变化的长期摄动影响。文中给出了太阳日多方指数ｎ＝３的模型由于自转、扁度和内部密度分布等因素对行星轨道要素变化的长期摄动效应的理论结果。研究结果表明：行星轨道要素除长轴、偏心率和轨道倾角不受长期摄动外升交点经度、近日点经度以及平近点角均受长期摄动的影响。最后利用理论结果对行星轨道要素的长期投动效应做了数值计算，数值结果在表１中给出。     

Dynamics of Planetary Satellites in the Solar System

After the discovery of a huge number of satellites around Jupiter, Saturn, and Uranus, it is necessary to collect together information about all of the planetary satellite systems and to define the possible classification of objects and types of their motion. We give physical parameters of the satellites: their masses, sizes, apparent magnitudes in opposition, and geometrical albedos. We present some of the orbital quantities that characterize the orbits, their shapes and orientation in space, as well as data on the rotation of satellites. The emphasis is on the peculiarities of their motion—the forces acting on them, the main orbital perturbations, and the influence of commensurabilities in the mean motions of satellites. We list references to the main theories of their motion.     

The onset of condensation: Association reactions

Association reactions are important in a variety of astronomical regions for increasing molecular complexity and as a prelude to condensation. Theories are presented for the two mechanisms — three-body and radiative — by which association reactions occur in the gas phase. The theory for three-body association is in satisfactory agreement with a variety of laboratory studies. This agreement lends confidence to the idea that both theories can be utilized for association reactions of astrophysical importance.Invited contribution to the Proceedings of a Workshop onThermodynamics and Kinetics of Dust Formation in the Space Medium held at the Lunar and Planetary Institute, Houston, 6–8 September, 1978.     

Possible effects of internal rotation in low-mass stars

The influence of internal rotation on the evolution of a 0.85M star is investigated by the construction of model sequences. Rotation is treated by a simple one-dimensional approximation. The calculations assume solid-body rotation on the zero-age Main Sequence, followed by conservation of angular momentum in shells. The 4 cases considered have the initial angular velocities 0,2×10^–4, 6×10^–4, and 8×10^–4/sec. All cases but the last are followed to helium ignition. Compared with the non-rotating case, the rotating models are older at Main-Sequence turnoff, develop fast-spinning central regions on the red-giant branch, and ignite helium at higher surface luminosities and at larger helium-core masses. The increases in the last two quantities are roughly proportional to the square of the initial angular velocity.The 6×10^–4 case is followed through the helium core flash to the zero-age horizontal branch. Under the assumption of spherical symmetry, the non-central ignition of helium leads to a sequence of flashes of decreasing amplitude occurring progressively closer to the center. The flashes are weaker than those encountered in previous studies and do not produce mixing.