On the relation of dynamics to statistical mechanics

A new conceptual framework for the foundations of statistical mechanics starting from dynamics is presented. It is based on the classification and the study of invariants in terms of the concepts of our formulation of non-equilibrium statistical mechanics. A central role is played by thecollision operator. The asymptotic behaviour of a class of states is determined by the collisional invariants independently of the ergodicity of the system. For this class of states we have an approach to thermodynamical equilibrium. We discuss the existence of classes of states which approach equilibrium. The complex microstructure of the phase space, as expressed by the weak stability concept which was introduced by Moser and others, plays here an essential role. The formalism that we develop is meaningful whenever the “dissipativity condition” for the collision operator is satisfied. Assuming the possibility of a weak coupling approximation, this is in fact true whenever Poincaré's theorem on the nonexistence of uniform invariants holds. In this respect, our formalism applies to few body problems and no transition to the thermodynamic limit is required. Our approach leads naturally to a ‘classical theory of measurement’. In particular a precise meaning can now be given to ‘thermodynamic variables’ or to ‘macrovariables’ corresponding to a measurement in classical dynamics.     

A new class of relativistic stellar models

Einstein field equations for a static and spherically symmetric perfect fluid are considered. A formulation given by Patiño and Rago is used to obtain a class of nine solutions, two of them are Tolman solutions I, IV and the remaining seven are new. The solutions are the correct ones corresponding to expressions derived by Patiño and Rago which have been shown by Knutsen to be incorrect. Similar to Tolman solution IV each of the new solutions satisfies energy conditions inside a sphere in some range of two independent parameters. Besides, each solution could be matched to the exterior Schwarzschild solution at a boundary where the pressure vanishes and thus the solutions constitute a class of new physically reasonable stellar models.     

The distinctive feature of relativistic stellar systems

In this paper we adopt the method of relativistic fluid dynamics to examine the number density distribution of stars around a massive black hole in the core of stellar clusters. We obtain extensive results,n(r) r ^–a, 3/2a9/2, which include, respectively, then(r) r ^–7/4 power law obtained by Bahcall and Wolf and then(r) r ^–9/4 power law by Peebles. Sincen(r) is not an observable quantity for star clusters, we also consider general relativity effects, i.e., the consequence of the bending of light, in calculating the projected density of stars in such a system. As an example we employ a massive black hole 10³ M inlaid in the center of a globular cluster and calculate various projected densities of stars. The results show that cusp construction occurs in all cases unless the central black hole massM=0, and the polytropic index does not affect at all the position of the capture radiusr _a. The obvious differences in the surface density is only embodied in the interior of the capture radius. At the outer regions of the core, the surface density of stars declines rapidly with ar ^–5 power law in all cases. These results can be applied to cases of unequal-mass and non-steady state.     

General relativistic rotational properties of stellar models

In this paper we give general relativistic expressions for the angular momentum and rotational kinetic energy of slowly rotating stars. These expressions contain contributions from the presure, gravitational red shift, and Doppler shift, and the motion of inertial frames. These contributions are not negligible, e.g., there are stable neutron star models for which the angular velocity of inertial frames at the center is about 70% the angular velocity of the star. These expressions are useful in the study of pulsars if pulsars are rotating neutron stars.     

Relativistic collapse of stellar clusters after loss of stability

The dynamics of spherically-symmetrical relativistic stellar clusters with initial truncated Maxwell distributions and initial self-similar distribution of stars is investigated using integral method of calculation. It is found that the main part of the clusters collapses rapidly. During long period after that the main part of remaining mass collapses and the most rapid members of the cluster run away.     

A relativistic model of stellar objects with core-crust-envelope division

In this work, we present a cogent and physically well-behaved solution for neutron stars envisaged with a core layer having quark matter satisfying the MIT-bag equation of state(Eo S), meso layer with Bose-Einstein condensate(BEC) matter satisfying modified BEC Eo S and an envelope having neutron fluid and Coulomb liquids satisfying quadratic Eo S. All the required physical and geometrical parameters like gravitational potentials, pressures, radial velocity, anisotropy, adiabatic index, mass function, compactification factor, and gravitational and surface redshift functions show a feasible trend and are continuous with smooth variation throughout the interior and across the regions of the star.Further, causality condition, energy conditions, static stability criterion(using Tolman-OppenheimerVolkoff equation) and Herrera cracking stability criterion are met throughout the star. The approach seems to be resulting in more realistic and accurate modeling of stellar objects, particularly realized by us for X-ray binary stars 4 U 1608–52(M = 1.7 M_⊙, R = 9.5 km) and SAX J1808.4–3658(M = 1.2 M_⊙,R = 7.2 km). Furthermore, we have ascertained that the continuity of the stability factor in all three regions of the stars demand a smaller core. As the core region of the star increases, the stability factor becomes discontinuous at all the interfaces inside the star.     

On the jubilee of academician V. A. Ambartsumyan statistical mechanics of stellar systems: From Ambartsumyan onward

The work of V. A. Ambartsumyan on stellar kinematics and dynamics is analyzed briefly. Its significance in the discussions of cosmogony during the 1930's is emphasized, especially as related to the victory of the “short” scale for galactic evolution. The theory of the destruction of star systems through evaporation of stars, which was founded by Ambartsumyan, is discussed. Ambartsumyan's theory of the equilibrium and evolution of binary stars in the galactic field and its subsequent development are outlined briefly.     

Relativistic dynamics of expanding sources

Given a relativistically expanding source, the transformation of the emission spectrum from the comoving frame of the source to the observer's frame is calculated. In particular, we deal with two special cases, that of a uniformly expanding sphere and that of a cylinder. In each case we consider a power-law spectrum with two distinct regions — the optically thick and thin parts — and transform those regions separately. The results are discussed, and a generalized procedure for transforming a spectrum of unspecified form is outlined. This procedure takes into account any bulk motion by the source as well as expansion. Finally, applications of this procedure to astrophysical sources are briefly discussed.     

The classification of stellar spectra using statistical image moments

A modeling of the response of the 1549 CIV emission line profile in the spectrum of Seyfert galaxy NGC 5548 to a sharp continuum drop which happened in 1989 is carried out on the basis of observational data taken from the literature. A more rapid response of the red wing of CIV profile suggesting the accretion of CIV-emitting clouds in the broad-line region (BLR) is reproduced. It is shown that the accretion velocity near the central object is consistent with a free fall law whereas at large distances from the center it varies approximately asr ^–2. The best agreement between observed CIV profile changes and modeled ones is obtained when the mass of the central objectM 1.2 × 10⁸ M . The first approximation of the BLR geometry is also found: the inflow takes place only outside two wide cones with half-opening angle 65° which are disposed like a sand-glass and filled by outflowing material. The angle between the axis of such biconical structure and our line of sight is found to be small ( 9°). It is shown that the outer radius of an accretion disk around the central object does not exceed ~ I light-day (i.e., < 100 Schwarzschild radii), and the outer radius of CIV-emitting BLR is 14 light-days. The nature of a broad blue excess in the CIV profile which did not respond to the continuum change is discussed.     

The dynamics of head-on collisions of spherical stellar systems

Energy changes in a head-on collision between two unequal Plummer model stellar systems (galaxies) are studied analytically under the impulsive approximation. The variation of the disruptive effects within and the mass escape from systems widely differing in mass and scalelength ratios are determined and some physical implications regarding the dynamical stability of the systems undergoing head-on collisions are indicated. It is found that if two systems differ considerably in size, both systems generally survive the collision if (i) the mass of the bigger is greater than about six times the mass of the smaller and (ii) the density of the smaller is more than about twenty-five times the entity of the bigger system, when the velocity at minimum separation is equal to the parabolic velocity of escape.     

Exact statistical mechanics of a one-dimensional self-gravitating system

The statistical mechanics of an isolated self-gravitating system consisting ofN uniform mass sheets is considered using both canonical and microcanonical ensembles. The one-particle distribution function is found in closed form. The limit for large numbers of sheets with fixed total mass and energy is taken and is shown to yield the isothermal solution of the Vlasov equation. The order of magnitude of the approach to Vlasov theory is found to be 0(1/N). Numerical results for spatial density and velocity distributions are given.     

Some applications of Jacobi dynamics to the stellar evolution

The stars in the Main Sequence are seen as a hierarchy of objects with different massesM and effective dynamical radiiR _eff=R/ given by the stellar radii and the coefficients for the inner structure of the stars.As seen in a previous work (Paper I), during the lifetime in the Main SequenceR _eff(t) remains a near invariant when compared to the variation in the time ofR(t) and (t).With such an effectiveR _eff one obtains the amounts of actionA _c(M), the effective densities _eff(M)=(M)³(M), the densities of action and of energy (or mean presures in the stellar interior)a _c(M),e _c(M), and the potential energiesE _p(M).The amounts of action areA _cM ^k withk1.87 for the M stars,k5/3 for the KGF stars, andk1.83 for the A and earlier stars, representing very simples conditions for the other dynamical parameters. For instancek5/3 means a near invariant effective density _eff for the KGF stars, while for such stars the mean densities and coefficients present the strongest variations with masses (M)M ^–1.81, (M)M^0.6.The cases for the M stars (e _c(M)M ^–1) and for the A and earlier stars (betweena _c(M)=constant and _eff(M)M ^–1) and also discussed. These conditions for the earlier stars also represent reasonable mean values for the whole stellar hierarchy in the range of masses 0.2M M25M .With all this, one can build dynamical HR diagrams withA _c(M), E_p(M), _eff M ^–p , etc., whose characteristics are analogous to these in the photometrical HR diagram. A comparison is made betweenA _c(M) from the models here and the HR diagram with the best known stars of luminosity classes IV, V, and white dwarfs.The comparison of the potential energiesE _p(M)M ^–p according to the stellar models used here and the observed frequency function (MM _–q (number of stars in a given interval of masses) from different authors suggests the possibility that the productE _p(M)(M) is a constant, but this must be confirmed with further studies of the function (M) and its fine structure.There are analogies between the formulation used here for the stellar hierarchy and other physical processes, for instance, in modified forms of the Kolmogorov law of turbulence and in the formulation used for the hierarchy of molecular clouds in gravitational equilibrium. Besides, the function of actionA _c(M) for the stars has analogous properties to the relations of angular momenta and massesJ(M) for different types of objects. The cosmological implications of all this are discussed.     

A nonlinear theory of stellar wind with allowance for the influence of relativistic particles

We solve the nonlinear problem of the dynamics of a steady-state, spherically symmetric stellar wind by taking into account particle acceleration to relativistic energies near the shock front. The particles are assumed to be accelerated through the Fermi mechanism, interacting with stellar-wind turbulence and crossing many times the shock front that separates the supersonic and subsonic stellar-wind regions. We take into account the influence of the accelerated particles on hydrodynamic plasma-flow parameters. Our method allows all hydrodynamic parameters of the shock front and plasma in the supersonic region to be determined in a self-consistent way and the accelerated-particle energy spectrum to be calculated. Our numerical and analytic calculations show that the plasma compression ratio at the shock front increases compared to the case where there are no relativistic particles and that the velocity profile in the supersonic region acquires a characteristic kink. The shape of the energy spectrum for the accelerated particles and their pressure near the front are essentially determined by the presumed dependence of the diffusion coefficient on particle energy, which, in turn, depends on the scale distribution of turbulent pulsations and other stellar-wind inhomogeneities.     

The importance of wide-system studies for stellar evolution and galactic dynamics

Wide pairs and wide multiple systems have been too much neglected during many years by visual double star astronomers with the argument that only close visual pairs (short periods) may lead to mass-determinations in a relatively short time interval. But mass-determination should not be considered as the only interest of double star astronomy, even if it is of a fundamental nature.Today, it appears that researches on the origin and the evolution of the wide systems are urgently wanted, not only for the understanding of the evolution of the stellar medium, but also for a better knowledge of galactic dynamics. Some examples are given.Presently, the main task for double star specialists will be an important improvement in double and multiple star census for all kind of systems: close, medium, and wide. The Hipparcos satellite will probably add some important informations in that respect but ground-based observations also remain of the highest importance (radial velocities, photometry, astrometry, etc.).Paper presented at the Lembang-Bamberg IAU Colloquium No. 80 on Double Stars: Physical Properties and Generic Relations, held at Bandung, Indonesia, 3–7 June, 1983.     

Effects of a clumpy circumstellar medium on the dynamics of stellar winds

Mass pickup from clumps resulting from previous mass loss phases by a fast stellar wind can substantially modify its behaviour, even leading to a highly supersonic wind becoming transonic before leaving the mass-loading region. The observational consequences of mass-loading on these astrophysical flows is discussed.