The physical properties of an analytic model for a relativistic star

An exact solution of Einstein's equations for a static isentropic perfect fluid sphere is examined in detail. The analysis yields a strong indication that the model isstable with respect to infinitesimal radial pulsations. This means that the temperature is decreasing outwards. We prove that the adiabatic speed of sound is everywhere less than the speed of light if and only if the radius of the sphere is larger than 1.61 times its Schwarzschild radius. We further show that the strong energy condition is fulfilled everywhere if and only if the radius is larger than 1.76 times the Schwarzschild radius. The necessary and sufficient condition for the speed of sound to be decreasing outwards is given, and we find that this criterion is fulfilled if the fluid is causal. Taking the values of the pressure and the density to be somewhere given by the maximum values from Baymet al.'s equation of state, i.e., ₀=5.1×10¹⁴ g cm^–3 andp ₀=7.4×10³³ dyne cm^–2, we find the maximum mass of the fluid sphere to be 2.5 solar masses.Dedicated to the memory of the late George Cunliffe McVittie (1904–1988).     

Radiative transfer equation in spherically-symmetric non-scattering media

We solved the equation of radiative transfer in spherically-symmetric shells with arbitrary internal sources. We integrated the equation of transfer on the discrete grid of angle and radius given by [_j–1, _j] [r_i–1, r_i]. The size in the angle coordinates is determined by the roots of a quadrature formula where as the size in the radial coordinate is determined by the non-negativity of the reflection and transmission operators. We considered two cases of variation of the Planck function. (1) Constant throughout the medium and (2) varying as 1/r ². We find that in the inner shells, the radiation directed toward the centre of the sphere is more than that directed away from the centre of the sphere. In the outer shells the converse is true.     

The formation of a ring galaxy during a head-on collision between a disk and a spherical galaxy

The tidal force effects of a spherical galaxy passing head-on through a disk galaxy have been studied at various regions of the disk galaxy and for various orientations of the disk galaxy with respect to the direction of relative motion of the two galaxies. The density distribution of the disk galaxy is taken to be, (r)=_ce^–4r/R , where _c is the central density andR is the radius of the disk. The density distribution of the spherical galaxy is taken to be that of a oolytrope of indexn=4. It is found that as a result of the collision, through the central parts and the outer parts of the disk galaxy remain intact, the region in between these two regions disrupts. Thus a ring galaxy with a nucleus embedded in the ring-i.e., a ring galaxy of the RN-type, is formed.     

Possible evidence for a massive black hole in a quasar with double redshift systems

With the aid of a simple black hole model of quasars we have found that the majority of the distinguishable emission lines in the spectrum of the quasar 1604+179 can be assigned to two redshift systems,z _r =3.712 andz _b =2.701. The appearance of double emission redshifts means that this quasar might be a massive black hole (of mass 10⁸ M M10¹¹ M ) with a ring-like emission line region (of radius 1 light-day r ₀1 light-year) in its accretion disk.     

Correlation Properties of a Distribution of Objects from the PSCz Survey

The properties of a sample of IR galaxies from the Point Source Catalog z (PSCz) Survey were investigated using the correlation gamma function [(r) and ^*(r)]. The results with different volume-limited subsamples indicate that the regions of strong correlations (a power-law decrease in density with distance with an exponent ₁ 1) are limited to a scale of 10-15 Mpc. A break is present at this scale in the dependences of log() on log(r) and of log(^*) on log(r). Such a break is also observed for various other samples of galaxies and clusters (at different scales). After the break the density dependence changes to another regime, corresponding to a fully uniform distribution for bright galaxies in the northern Galactic hemisphere. For some subsamples of the southern and northern hemispheres the latter regime corresponds to some decrease in density with distance. Indications of significant differences between the distributions of objects in the southern and northern hemispheres are obtained. It is shown that the section of the gamma function after the break, even when its extent is small, is a significant indicator of actual correlation properties of the distributions. The results of the analysis are in good agreement, on the whole, with preceding studies of the PSCz survey.     

The number-density of stars in a spherically-symmetric system containing a massive black hole at its center

An object with a compact core is a configuration in various astrophysical problems. In this paper, considering general relativity, the number densities of stars in a spherically-symmetric system containing a massive black hole in its center has been obtained forv=4/3,5/3, and the non-relativistic case. The results obtained by Peebles (1972), Bahcall and Wolf (1976), and that obtained for an isothermal sphere case are discussed. Taking into account the stability of the system, it is found to be more rational to usen(r) r ^–7/4 to express the number density of stars when they are near the massive central black hole.     

Structure of the expanding atmosphere of P Cygni

With the aid of the spectra taken in the years 1959–1968, a physical analysis of the atmosphere of P Cygni has been carried out and the motions of the atmosphere have been studied. The variations of radial velocities, the velocity progressions of Balmer and Hei lines, the high rate of mass loss (2×10^–5 M yr^–1), the features of the observed line profiles, especially that of H-K lines of Caii andD ₁-D ₂ lines of Nai confirm the conclusion of Van Blerkom (1978), concerning the assumption of an accelerating atmosphere for P Cygni. The electron density variation with the radius seems to ben _e r ^–5/2, with an average value of 7×10¹¹cm^–3 at the lower boundary of the atmosphere.In order to explain the two absorption components of observed lines, an atmospheric model based on the assumption of three envelopes, two of which accelerate gradually with two different velocity laws (up to 11.2r _c ), and the third of which accelerates rapidly with a standard velocity law (beyond 11.2r _c ) has been developed. From this model and the observed profiles, the geometrical thicknesses of the line-forming regions of H, H, H, and H are derived.The observations were obtained at Haute Provence Observatory (CNRS).     

Gas and dust release from comet West (1976VI) nuclear fragments

Emission fluxes of CN, G₂ and C₃ carbon-bearing molecular species observed in the spectra of the nuclear fragment A and D of comet West (1976VI) are analysed in the framework of Haser's two-component model with radial outflow symmetry. Gas production rates have been derived and their ratios obtained. We find these vary as Q r ⁿ (where 1.0r2.6 is the heliocentric distance in AU) with photometric parametern = -4.11 (CN), -2.68 (C₂), -1.89 (C₃) andQ _x /Q _y r ⁿ withn = 2.22 (C₃/CN), 1.42 (C₂/CN), -0.79 (C₂/C₃). The gas-to-dust ratio, dust production rates in arbitrary units, total number of dust particles in the field of view and an effective radius R_eff 2.0 km for the principal nuclear fragmentA is also estimated.     

Determination of the electromagnetic field produced by a magnetic oblique-rotator

The electromagnetic field produced by a magnetic dipole moment, , which is rotating obliquely surrounded by a corotating plasma sphere, is investigated. This corotating-plasma approximation has the same order of accuracy as the force-free one but has somewhat different physical implications. In the former the effect of non-electromagnetic forces such as the inertial force are included, though in somewhat artificial manner, as a departure from the strict MHD condition and this fact seems to guarantee the existence of physical solutions.Analogous to the relativistic force-free equation, a set of two differential equations (the corotation equation) are derived for the scalar functions associated with the electric and magnetic fields. A self-consistent solution of these equations is given and it is shown that this solution has no singularity, in spite of apparent divergence in the formal solution, on the light cylinder. It is concluded from this solution that, even in the extreme case of the largest possible corotation-radius (i.e.b=r _L , wherer _L is the light radius), the existence of a corotating plasma does not alter the field structure drastically from the vacuum case. It is also suggested through this treatment that inclusion of the inertial term in generalized Ohm's law might be essential in considering the centrifugal-wind problem.     

An integrable case of a rotational motion analogous to that of Lagrange and Poisson for a gyrostat in a Newtonian force field

The scope of the present paper is to provide analytic solutions to the problem of the attitude evolution of a symmetric gyrostat about a fixed point in a central Newtonian force field when the potential function isV ⁽²⁾.We assume that the center of mass and the gyrostatic moment are on the axis of symmetry and that the initial conditions are the following: (t ₀)=₀, (t ₀)=₀, (t ₀)=(t ₀)=₀, ₁(t ₀)=0, ₂(t ₀)=0 and ₃(t ₀)= ₃ ⁰ .The problem is integrated when the third component of the total angular momentum is different from zero (B ₁ 0). There now appear equilibrium solutions that did not exist in the caseB ₁=0, which can be determined in function of the value ofl ₃ ^r (the third component of the gyrostatic momentum).The possible types of solutions (elliptic, trigonometric, stationary) depend upon the nature of the roots of the functiong(u). The solutions for Euler angles are given in terms of functions of the timet. If we cancel the third component of the gyrostatic momentum (l ₃ ^r =0), the obtained solutions are valid for rigid bodies.     

Making Sense of Sunspot Decay. I. Parabolic Decay Law and Gnevyshev–Waldmeier Relation

In a statistical study of the decay of individual sunspots based on DPR data we find that the mean instantaneous area decay rate is related to the spot radius r_o and the maximum radius r_o as D = C_D r/r_o, C_D = 32.0±0.26 MSH day ^-1. This implies that sunspots on the mean follow a parabolic decay law; the traditional linear decay law is excluded by the data. The validity of the Gnevyshev–Waldmeier relationship between the maximum area A ₀ and lifetime T of a spot group, A₀/T 10 MSH day^-1, is also demonstrated for individual sunspots. No evidence is found for a supposed supergranular quantization of sunspot areas. Our results strongly support the recent turbulent erosion model of sunspot decay while all other models are excluded.     

A semiquantitative general-relativistic model of quasar Markarian 205 interpreted as a massive black hole ejected from NGC 4319

In 1979 I developed a special-relativistic analysis explaining the discrepancy of observed redshifts of spiral NGC 4319 and its companion quasar Markarian 205 by considering besides the Lorentz time dilatation also the gravitational redshift due to the gravitational field of Markarian 205 interpreted in terms of accretion of mass onto a black hole ejected from NGC 4319. In the present paper, a general-relativistic analysis is given. Numerical results of the special and general theories do not differ from each other significantly and admit the conclusion that the radius,r, of the radiating region of Markarian 205 is of the order of the tidal radius of black hole. Several models for various values of the ratio ofr to the Schwarzschild radius,r _s, are constructed. Models with 8.5r/r _s8.7 seem to be most realistic. It becomes clear that the interpretation of quasars in terms of huge black holes accreting stars can explain, in principle, the observed redshifts of quasars ejected from parent galaxies.     

A complete model of the propagation of solar-flare cosmic rays

A two-stage model of the propagation of 1–50 MeV solar-flare cosmic rays is presented. The first stage consists of a thin spherical shell of radius r _a near the Sun which feeds particles into interplanetary space (the second stage) where they propagate along the Archimedean mean interplanetary magnetic field under the influences of anisotropic diffusion, convection, and energy changes. To calculate the time dependence at a fixed point in space, account is taken of the corotation of flux tubes past the observer.It is shown that the well-known east-west effect of the time-to-maximum cannot be obtained if the injection from the first stage is impulsive and thus a time and longitude dependent release for the second stage is essential. This is achieved by treating the first stage as a thin, spherical, diffusing shell of radius r _a with diffusion coefficient _s, from which particles leak into interplanetary space at a rate determined by the leakage coefficient .With this model we are able to reproduce simultaneously four principal features of solar events observed at r = 1 AU: (i) the east-west effect, i.e. the time-to-maximum as a function of flare longitude; (ii) the three phases of the anisotropy vector variation; (iii) the time-to-convective-phase as a function of flare longitude; and (iv) the longitudinal distribution of the differential intensity. Our best estimates of the parameters of the near-Sun propagation are that 0.01 hr^–1 _s/r _a ² 0.02 hr^–1 and 1/15 hr^–1 1/10 hr^–1. For the interplanetary propagation we estimate /V - 1.2AU with , the effective cosmic-ray diffusion coefficient and V, the solar-wind speed.     

An extended investigation of Helios 1 and 2 observations: The interplanetary magnetic field between 0.3 and 1 AU

Helios-1 and 2 spacecraft allowed a detailed investigation of the radial dependence of the interplanetary magnetic field components between 0.3 and 1 AU. The behaviour of the radial component B _ris in a very good agreement with Parker's model (B _r r ^-2) and the azimuthal component B also shows a radial dependence which is close to theoretical predictions (B r ^-1). Experimental results for the normal component B and for the field magnitude B are consistent with those from previous investigations. The relative amplitude of the directional fluctuations with periods less than 12 hr is essentially independent of heliocentric distance, while their power decreases approximately as r ^–3 without any appreciable difference between higher and lower velocity regimes.Also at Laboratorio Plasma nello Spazio, CNR, Frascati.     

One-armed oscillations of a non-selfgravitating polytropic disk

One-armedglobal oscillations in a non-selfgravitating polytropic disk rotating around a star are investigated. The unperturbed disk is axisymmetric, geometrically thin, and extends infinitely in the radial direction keeping its thickness constant. Perturbations considered are inviscid and adiabatic. It is found that there are one-armed retrograde wave modes which are trapped in an inner region of the disk. The eignefrequency of the lowest order mode is given by _K(r _s)(z ₀/r _s)₂, wherer _s is the radius of the central star,z ₀ is the half-thickness of the disk, and _K(r _s) is the Keplerian angular frequency at the surface of the star.Paper presented at the IAU Third Asian-Pacific Regional Meeting, held in Kyoto, Japan, between 30 September–6 October, 1984.     

Time-dependent chemistry in expanding circumstellar envelopes

The rate equations of a restricted set of gas-phase chemical reactions occuring in an expanding circumstellar envelope are integrated numerically on the assumption that no chemical evolution has occurred in the stellar atmosphere. Abundances of all species are found to peak at a time on the order ofr ₀/u ₀, wherer ₀ is the initial radius andu ₀ the expansion velocity. After this time geometrical dilution dominates. For an initial density of 10⁸ cm^–3, on the order of 1% of hydrogen is converted to H₂, and CH and CO have comparable densities of 10^–6 relative to H, OH and O₂ remain very low in abundance. For higher initial densities, H and H₂ are more nearly comparable, and nearly all carbon is in CO, CH, OH and O₂ remain low in abundance. The relevance of these results to M giants and other objects is discussed.     

Possible effects of anisotropy ofG on celestial orbits

Will (1971) has discussed a possible anisotropy in the gravitational constantG. Suppose that the attractive gravitational force between two particles of massesm ₁ andm ₂ is given by the usual expressionF=–Gm ₁ m ₂ r/r ³, wherer is the separation vector. Ifc is the velocity of light in vacuo and if 1 _r r/r, he expresses the anisotropy byG=G [1+(v·1 r/c)²], whereG is a constant,v is identified practically as the velocity of the Sun around the galaxy, and 1. Will's suggestion is to look for such an effect in the laboratory.The purpose of the present paper is to look for such an effect in the solar system, wherem ₁ andm ₂ become the masses of the Sun and a planet or of the Earth and the Moon. For simplicity I consider only those planets whose orbits are close to the ecliptic, so that the angle betweenv and the plane of the ecliptic is about 59°.With the above force, the resulting two-body problem is completely solvable. The results are these. If =1, there is an increase in mean motion of 7 parts in 10⁸, a periodic fluctuation in true longitude with period half that of the orbit and amplitude ranging possibly from 0.01 to 0.02, and periodic fluctuations in the radius vector, with period also one half that for the orbit. The amplitudes are: 2.7 km for Mercury, 5.1 km for Venus, 7.0 km for Mars, 18 m for the Moon about the Earth, and 28 cm for a close artificial satellite with inclination 23°. The more conservative estimate <0.0115 would reduce these values by the factor 70.     

Further notes on the associated alpha-functions and related integrals in the theory of the light changes of eclipsing variables

With a view to furthering the theory of the light changes of eclipsing variables, developed before systematically by Z. Kopal, this paper presents a number of new (and computable) expressions for the associated alpha-function _n ⁰ ,(r₁,r₂,) (and also for its partial derivatives), where _n ⁰ ,(r₁,r₂,) represents the fractional loss of light suffered by an eclipse of a circular disc of fractional radiusr ₁ (and darkened at the limb to thenth degree) by an opaque disc of radiusr ₂, with their centres separated by a fractional (projected) distance , provided that the transparency of the occulting disc increases with the angle of foreshortening in the same manner as the limb-darkening of the eclipsed star (that is, when the transparency functiong(, ) of the second aperture is given by Equation (4) below). Many of the explicit expressions derived here are valid for any type of eclipse, occultation or transit, regardless of whetherr ₁>r ₂ orr ₁<r ₂. and for any degreen of the adopted law of limb-darkening. It is also pointed out how some of the results obtained in this paper are related to the various representations given earlier in the literature for the case =0.     

Dust emission from comets at large heliocentric distances

Possible mechanisms for the production of the extended dust coma observed in comet Bowell (1980b) at the large heliocentric distance of 7.17 AU are considered. It is concluded that a plausible mechanism is electrostatic blow-off of fine, loose dust from an electrically charged H₂O dominated nucleus, as recently proposed by Mendiset al. (1981). Of all the other processes considered, dust entrainment by a chemical species more volatile than H₂O is considered the most plausible. However, the non-observation of prominent CO⁺ lines of theA ²-X ² band, seems to mitigate against a CO dominated nucleus, particularly if its radius is 1 km. A CO₂ dominated nucleus, however, cannot be ruled out by this non-observation unless its radius is considerably larger ( 5 km). The electrostatic blow-off process predicts that the dust grains in the coma of comet Bowell at 7.17 AU must be very small (r _g0.4 ), with a total mass 4×10⁸ g, and a spatial extension 10⁵ km, which is consistent with the observations (observed coma diameter 6×10⁴ km). Both the size and spatial distribution of the dust given by electrostatic blow-off and the gas entrainment processes are shown to be different, and this would be a way of discriminating between the two processes. However, the present observations do not have the resolution to do so.Department of Physics.Department of Electrical Engineering and Computer Sciences.     

Green's theorem and Green's functions for the steady-state cosmic-ray equation of transport

Green's Theorem is developed for the spherically-symmetric steady-state cosmic-ray equation of transport in interplanetary space. By means of it the momentum distribution functionF _o(r,p), (r=heliocentric distance,p=momentum) can be determined in a regionr _arr_bwhen a source is specified throughout the region and the momentum spectrum is specified on the boundaries atr _a andr _b . Evaluation requires a knowledge of the Green's function which corresponds to the solution for monoenergetic particles released at heliocentric radiusr _o , Examples of Green's functions are given for the caser _a =0,r _b = and derived for the cases of finiter _a andr _b . The diffusion coefficient is assumed of the form = _o(p)r ^b . The treatment systematizes the development of all analytic solutions for steady-state solar and galactic cosmic-ray propagation and previous solutions form a subset of the present solutions.