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.     

Co-Orbital Motion with Slowly Varying Parameters

We consider the dynamics of a test particle co-orbital with a satellite of mass m _s which revolves around a planet of mass M ₀ m _s with a mean motion n _s and semi-major axis a _s. We study the long term evolution of the particle motion under slow variations of (1) the mass of the primary, M ₀, (2) the mass of the satellite, m _s and (3) the specific angular momentum of the satellite J _s. The particle is not restricted to small harmonic oscillations near L ₄ or L ₅, and may have any libration amplitude on tadpole or horseshoe orbits. In a first step, no torque is applied to the particle, so that its motion is described by a Hamiltonian with slowly varying parameters. We show that the torque applied to the satellite, as measured by _s = j_s/(n _s J _s) induces an distortion of the phase space which is entirely described by an asymmetry coefficient = _s/, where = m _s/M. The adiabatic invariance of action implies furthermore that the long term evolution of the particle co-orbital motion depends only on the variation of m _s a _s with time. Applying a constant torque to the particle, as measured by _s = j_s/(n _s J _p) is then merely equivalent to replacing = _s/ by = (_s – _p)/. However, if the torque acting on the particle exhibits a radial gradient, then the action is no more conserved and the evolution of the particle orbit is no more controlled by m _s a _s only. We show that even mild torque gradients can dominate the orbital evolution of the particle, and eventually decide whether the latter will be pulled towards the stable equilibrium points L ₄ or L ₅, or driven away from them. Finally, we show that when the co-orbital bodies are two satellites with comparable masses m ₁ and m ₂, we can reduce the problem to that of a test particle co-orbital with a satellite of mass m ₁ + m ₂. This new problem has then parameters varying at rates which are combinations, with appropriate coefficients, of the changes suffered by each satellite.     

A new examination of absorption line density in the Lα forest of QSOs

The distribution of the absorption lines in the L forest of QSOs is re-examined in relation to both absorption and emission redshifts by analyzing a sample consisting of 20 QSO L forests.On the one hand, the -value in the relationN(z _abs)=N ₀(1+z _abs) is re-estimated by applying the maximum likelihood (ML) method to all the L forests in our sample as a whole as well as to the forests for individual QSOs. The global estimated is 2.20±0.28 and both the quasars with highest and lowestz _em do not individually have a significant effect on it. A non-parametric test on the estimate of from individual QSOs shows that there is no significant difference between the numbers of positive and negative -values. This does not, therefore, favour the idea that for individual QSOsN(z _abs) decreases with increasingz _abs.On the other hand, the possible correlation between the absorption line density and the emission redshift of the QSO itself, which was found by Bianet al. (1986, 1988) and re-discovered independently by Tytler (1987), is shown once again not only by the largerz _em QSOs having larger mean density, but more importantly, at a rather high confidence level, by the fact that, at the same values ofz _abs, QSOs with largerz _em possess a greater number of L absorption lines.A discussion of the inverse effect suggests that it is still hard to say whether this effect really exists.     

The X-ray properties of quasars determined from an iterative multiple regression analysis with censored data

We adopt the iterative least-squares method of Schmee and Hahn (1979) and extend it to the multiple regression case to investigate the dependence of the X-ray luminositíesL _x of quasars on their optical liminositiesL ₀ and redshiftsz. The method has several advantages, one of which is that we do not need the strong assumption that the values of the optical-X-ray index _0x of the detected and nondetected quasars should fall into the same range. We have found that correlations of _0x and log(1+z) or the look back time (z) are shown, but these are not firmly established due to the uncertainty present in the regression parameter. On the other hand, it is impossible for _0x to be independent ofL ₀. We have also shown thatL _x is less than proportional toL ₀.     

The seven components of Hα and the 9873 MHz line

Under conditions pertaining to the middle chromosphere, the nlj-substates of the n = 2 and n = 3 states of hydrogen are found to be populated proportionally to their degree of degeneracy. Thus, the non-LTE formation of the H line is not expected to be influenced by some exotic excitation conditions in one of its seven components. The overpopulation of the 2S _1/2 substate relative to the 2P _3/2 substate does not depend upon details of the radiative transfer in H, so that it rests wholly on the transfer in L (Milkey and Mihalas, 1973). One-component plane parallel models of the chromosphere thus further predict that the 2S _1/2-2P _3/2, 9873 MHz (3.04 cm) line will not be observable in the radio-spectrum of the quiet Sun.Astronomische Mitteilungen der Eidgenössischen Sternwarte Zürich, Nr. 367.     

Rapidly rotating supermassive stars in the first post-Newtonian approximation to general relativity

The structure and stability of rapidly uniformly rotating supermassive stars is investigated using the full post-Newtonian equations of hydrodynamics. The standard model of a supermassive star, a polytrope of index three, is adopted. All rotation terms up to and including those of order ⁴, where is the angular velocity, are retained. The effects of rotation and post-Newtonian gravitation on the classical configuration are explicitly evaluated and shown to be very small. The dynamical stability of the model is treated by using the binding energy approach. The most massive objects are found to be dynamically unstable when =1/c ².p _c / _c 2.2 × 10^–3, wherep _c and _c are the central pressure and density, respectively. Hence, the higher-order terms considered in this analysis do not appreciably alter the previously known stability limits.The maximum mass that can be stabilized by uniform rotation in the hydrogen-burning phase is found to be 2.9×10⁶ M , whereM is the solar mass. The corresponding nuclear-generated luminosity of 6×10⁴⁴ erg/sec^–1 is too small for the model to be applicable to the quasi-stellar objects. The maximum kinetic energy of a uniformly rotating supermassive star is found to be 3×10^–5 Mc ², whereM is the mass of the star. Masses in excess of 10¹⁰ M are required if an adequate store of kinetic energy is to be made available to a pulsar like QSO. However such large masses have rotation periods in excess of 100 yr and thus could not account for any short term periodic variability. It is concluded then that the uniformly rotating supermassive star does not provide a suitable base for a model of a QSO.     

Period distributions in pulsating and binary stars

We analyze the hypothesis of quantization in bands for the angular momenta of binary systems and for the maount of actionA _c in stable and pulsating stars. This parameter isA _c=Mv _eff R _eff, where the effective velocity corresponds to the kinetic energy in the stellar interior and the effective radius corresponds to the potential energyGM ²/R _eff. Analogous parameters can be defined for a pulsating star withm=M where is the rate of the massm participating in the oscillation to the total massM andv _osc,R _osc the effective velocity and oscillation radius.From an elementary dimensional analysis one has thetA _c (energy x time) (period)^1/3 independently ifA _c corresponds to the angular momentum in a binary system, or to the oscillation in a pulsating star or the inner energy and its time-scaleP _eff in a stable star.From evolving stellar models one has that P _effP _eff(solar)1.22 hr a near-invariant for the Main Sequence and for the range of masses 0.6M <M<1.6M .With this one can give scalesn _k=kn ₁ withk integers andn ₁=(P/P ₁)^1/3 withP ₁=P _eff1.22 hr. In these scales proportional toA _c, one sees that the periods in binary and pulsating stars are clustered in discrete unitsn ₁,n ₂,n ₃, etc.This can be seen in pulsating Scuti, Cephei, RR Lyrae, W Virginis, Cephei, semi-regular variables, and Miras and in binary stars as cataclysmic binaries, W Ursa Majoris, Algols, and Lyrae with the corresponding subgroups in all these materials. Phase functions (n _k) in RR Lyrae and Cephei are also associated with discrete levelsn _k.the suggested scenario is that the potential energies and the amounts of actionE _p(t), A_c(t) are indeed time-dependent, but the stars remain more time in determinated most proble states. The Main Sequence itself is an example of this. These most probable states in binary systems, or pulsating or stable stars, must be associated with velocities sub-multiplesc/ _F , given by the velocity of light and the fine structure constant.Additional tests for such a hypothesis are suggested when the sufficient amount of observational data are available. They can made with oscillation velocities in pulsating stars and velocity differences of pairs of galaxies.     

Thermalization of solar wind

Three kinetic equations describing the linear and non-linear wave-particle interaction for an anisotropic solar wind plasma have been developed. These equations have been solved numerically to find the variation inT /T with respect to time, whereT andT are the perpendicular and parallel temperatures with respect to the ambient magnetic field of the solar wind. For wave energy greater than a critical value (strong turbulence), non-linear wave-particle interactions are important but do not lead to thermalization. On the other hand, weak nonlinear interactions tend to increaseT /T , but make only a negligible contribution in the quantitative sense. Thus, only the linear wave-particle interaction remains as the significant contributer to the increase ofT /T .     

Genericity of the non-periodic solutions of the central force problem

We study the classical problem of two-dimensional motion of a particle in the field of a central force proportional to a real power of the distancer. for negative energy and (0, 2), each energy levelI _h is foliated by the invariant toriI _hc of constant angular momentumc and, by Liouville-Arnold's theorem, the flow on eachI _hc is conjugated to a linear flow of rotation number _h (c).A well-known result asserts that if we require _h (c) to be rational for every value ofh andc, the, must be equal to one (Kepler's problem). In this paper we prove that for almost every (0, 2) _h (c) is a non-constant continuous function ofc, for everyh<0. In particular, we deduce that motion under central potentials is generically non-periodic.Partially supported by CIRIT under grant No. EE88/2.     

Physical processes determining the chromospheric temperature distribution

Calculations performed with several models of the solar chromosphere support Ulmschneider's conclusion that relatively short period acoustic waves heat the low chromosphere in the region just above the temperature minimum. However, these same short period waves (10 period P80 s) are not able to maintain chromospheric temperatures at heights where _5000Å(normal) < 10^-6. The calculations also show that an earlier conjecture stating that the H₂ population might influence the non-LTE chromospheric H^- population is probably not correct, due to lower values of the ratio n _e/n _H inferred from more recent observations. Finally, the calculations support Athay's contention that the Cayrel mechanism alone cannot produce the observed temperature rise, because the magnitude of the radiative cooling in the lines is too great.     

Constraints on the gravitational constant from the observations of white dwarfs

Recently some authors have questioned whether Newton's law of gravitation is actually true on scales less than 1 km. The available constraints on the gravitational constant show that is laboratory valueG ₀ may differ from the value at infinityG by 40%. Long (1976) reported experimental evidence for departures from Newton's law. In this note it is shown that the difference betweenG ₀ andG modifies the mass-radius relation of degenerate stars. The observations of white dwarfs are consistent with the theory of stellar evolution only ifG ₀ differs fromG by not more than 10%. This estimate may be improved by a higher accuracy of observations.     

Existence of the continuations in theN-body problem

The method of obtaining the estimates of the maximalt-interval ( _–, ₊) on which the solution of theN-body problem exists and which is such that some fixed mutual distance (e. g. ₁₂) exceeds some fixed non-negative lower bound, for allt contained in ( _–, ₊), is considered. For given masses and initial data, the increasing sequences of the numbers _k , each of which provides the estimate ₊ > _k , are constructed. It appears that if ₊ = +, then .     

Evolution of relative helicity in active regions

The evolution dH _R/dt of relative helicity H _R provides a gauge-invariant measure of the helicity flow across the open surface S _o of an active region. With the incompressible approximation, reformulation of the evolution reveals that it is determined not only by the widely used cross-helicity h _mvp=A _pv contributed by the vector potential A _p of a reference potential field B _p, where v is the fluid velocity, but by another cross-helicity h _mvo=A _ov contributed by the vector potential A _o of the open field B _o in the region as well. Only under two conditions, (1) A _p=A (A is the transverse component of A _o), (2) v _z=0 (v _z is the longitudinal component of v) or A _z=0 (A _z is the longitudinal component of A _o), can h _mvo be merged into h _mvp to give the pioneering dH _R/dt equation shown in Equation (4) of Berger (1984). Results show that h _mvo originates from vh _o (h _o=A _oB _o is the helicity density of the open field) and should also be considered in dealing with the development of relative helicity in active regions. Finally, the equation to calculate dH _R/dt in active regions is synthesized and presented.     

Magnetic and thermal pressures in the solar wind

Explorer 34 solar wind data for the period June to December, 1967 show that(a) The magnetic pressure, P _BB ²/8, and thermal pressure,P _kn _p kTp+n kT+n _e kTe,are variable and positively correlated on a scale of 2 days, but (b) changes in P _b and P _k are anticorrelated on a scale 1 hr (0.01 AU). Thus, dynamical hydromagnetic processes (dv/dto) must occur on the mesoscale, but the solar wind tends to be in equilibrium(P _B+P _Kconstant) on a smaller scale, the microscale. The 3-hr averages show that the most probable value of P _{k/P B} is =1.0±0.1, which implies that the most probable state of the solar wind at 1 AU is not one of equipartition between the thermal energy and magnetic energy. The average total pressure for a given bulk speed(P(V)=P ^k+P _k+P _B) is essentially independent of V, implying that P is not determined by the heating or acceleration mechanisms of the solar wind; the average pressure is P=(2.9±1.5)×10^-10dyne/cm².     

Neutron stars and general equation of nuclear matter

It has been shown that the mass of neutron stars obtained from equations of state based on nuclear theory depend upon the number of baryons assembled in it but not on the type of interactions considered. On examining the behaviour of different equations of state based on nuclear theories, a simple polytropic equation of state,P = (K/N)(p –p _s)^N is proposed. The results obtained forN=1.75 cover the entire range of neutron star masses obtained from the equations of state based on nuclear theories and give a maximum mass of 2.8M . Depending upon various mechanisms for energy output the mass of Crab pulsar is estimated to range from 0.32M to 1.5M . The relation connecting the coordinate mass,M, and the rest mass,M ₀, may be written asM/M 0.93 (M ₀/M)^0.9.     

Mean free paths of energetic particles at very large heliodistances (Pioneer 11 at 20 AU)

Pioneer 11 magnetic field data at 20 AU are analysed by the computational method of Moussas, Quenby, and Webb (1975), Moussas and Quenby (1978), and Moussas, Quenby, and Valdes-Galicia (1982a, b) to obtain the parallel mean free path , and the diffusion coefficient parallel to the magnetic field line K . This method is the most appropriate for the mean free path calculation at large heliodistances since the alternative method which is based on fitting of energetic particle intensities cannot be easily and accurately be used because the association of energetic particles with their parent flares is not precise. The results show that the mean free path has values between 0.85 and 0.98 AU, linearly increasing with energy according to (T_kinetic) = + MT, where = 0.846 AU and M = 4.44 × 10 ^–5 AU MeV^–1 for energies between 10 MeV and 3 GeV for protons. These values of the parallel mean free path are much larger than the values estimated by previous studies up to 6 AU. The diffusion coefficient dependence upon energy follows a relation which simply reflects an almost constant mean free path and a linear dependence on the velocity of the particle, so that at 20 AU heliodistance K (T _kin) = K _{, 1 MeV}(T _kin)T _kinetic , with = 1/2. The distance dependence of the parallel diffusion mean free path follows a power law, (R) = _{, 1 AU} R , where is 1 ± 0.1. While the parallel diffusion coefficient obeys a power-law relation with heliodistance R, K (R, T _kin) = K _{, 1 AU}(T _kin)R , with = 1 ± 0.1. The radial diffusion coefficient of cosmic rays is not expected to strongly depend upon the parallel diffusion coefficient because the nominal magnetic field at these large heliodistances (20 AU) is almost perpendicular to the radial direction and the contribution of the diffusion coefficient perpendicular to the magnetic field is expected to play a dominant role. However, the actual garden hose angle varies drastically and for long time periods and hence the contribution of the diffusion parallel to the field may continue to be important for the small scale structure of intensity gradients.     

The mass-ratio distribution of spectroscopic binary stars

In order to determine the mass-ratio distribution of spectroscopic binary stars, the selection effects that govern the observations of this class of binary systems are investigated. The selection effects are modelled numerically and analytically. The results of the models are compared to the data inThe Eighth Catalogue of the Orbital Elements of Spectroscopic Binary Stars (DAO8) compiled by Battenet al. (1989). The investigations involve binary systems with Main-Sequence primary components only, in order to avoid confusion of evolutionary and selection effects.For single-lined spectroscopic binaries (SBI) it is found that the mass ratios (q=M _sec/M _prim) in general adhere to a distribution _q q ^-2 forq>q ₀, withq ₀=0.3. The observations are consistent with a distribution that is flat forq<q ₀. The turn-over value varies fromq ₀=0.3 for systems with B-type primaries, toq ₀=0.55 for systems with K-type primaries. The semi-major axesa ₁ are distributed according to _a (a ₁)a ₁ ^-a with an average value of _a =1.3. The power varies from _a =1.7 for systems with B-type primaries to _a =0 for systems with K-type primaries. The eccentricitiese of the orbits of SBI systems are distributed according to _e (e)e ^-1.For double-lined spectroscopic binary stars (SBII) it is found that the shape of theq-distribution, as derived from observations, is almost entirely determined by selection effects. It is shown that the distribution is compatible with theq-distribution found for SBI systems. A sub-sample, consisting of the SBII systems from DAO8 with magnitudesm _V 5 ^m , is less hampered by selection effects, and shows the same shape of theq-distribution as the SBI systems, at theq-interval (0.67, 1).It is estimated that 19–45% of the stars in the solar neighbourhood are spectroscopic binary systems.     

Nebulosities around QSOs

In this paper independent evidence in favour of the hypothesis proposed by Thakur and Sapre (1979) that a QSO consists of a bright central object embedded in an extended nebulosity has been presented. _U–B and _B–V , the spectral indices in (U-B) and (B-V) colours, have been calculated for a sample of 80 QSOs with redshiftz0.76. In Figure 1 = ( _B–V – _U–V ) has been plotted against = ( _B–V + _U–B )/2. In this figure the QSOs in which detectability of neubulosities has been predicted by Thakur and Sapre (1979) occupy a separate but adjacent part of the diagram as compared to those for which such a prediction has not been made. In Figure 2 (U-B)₀ has been plotted against (B-V)₀ for the same sample of 80 QSOs, where (U-B)₀ and (B-V)₀ are the (U-B) and (B-V) colours corrected for galactic extinction. In this plot also, a similar separation of the two classes of objects is discernible. Another empirical criterion—namely, (B-V)₀0.32 for the detectability of nebulosities around QSOs—has been suggested.     

Restricted quatum-mechanical three-body problems

On the basis of a perturbative procedure in which the eigenfunctions of a helium-like ion are expanded in the Hilbert space built up from the eigenfunctions of an electron in two fixed Coulomb charges,all asymptotic eigenfunctions are constructed for the Schrödinger equation of helium-like ions. If the nuclear chargeZ ₁ is not less than 2, then our asymptotic considerations clarify the singularities of the Schrödinger equation of a helium-like ion (atr ₁=0, ,r ₂=0, ,r ₁₂=0, ), while in the case ofZ ₁=1 (negative hydrogen ion)r ₂=0 will not be treated in this paper. The established order (inr ₂) of asymptotics at 0 or in an exceptional case the zeroth-order term of the functions (actually the coefficients of an expansion of the desired eigenfunctions) as one of the electron coordinates (r ₂, the distance of the two fixed Coulomb charges, a parameter of the set of the basic functions) enables us to classify the eigenfunctions of a helium-like ion. This classification resembles the classification scheme for one-electron configurations. The asymptotics forr ₂ indicate bounded, pseudobounded (auto-ionizing) and free states. (Doubly ionized continuum states are not discussed here.) The use of the asymptotic solutions is indicated for the complete solution of the problem which may be either numerical integration or a variational procedure.Neutral muonic helium is included in the discussion.     

Effect of perturbations in Coriolis and centrifugal forces on the stability of libration points in the restricted problem

The location and the stability of the libration points in the restricted problem have been studied when small perturbation and are given to the Coriolis and the centrifugal forces respectively. It is seen that the pointsL ₄ andL ₅ form nearly equilateral triangles with the primaries and the pointsL ₁,L ₂,L ₃ remain collinear. It is further observed that for the pointsL ₄ andL ₅, the range of stability increases or decreases depending upon whether the point (, ) lies in one or the other of the two parts in which the (, ) plane is divided by the line 36-19=0 and the stability of the collinear points is not influenced by the perturbations and they remain unstable.