Self-gravitating Mass Research Articles

An algorithm for computing the 2D structure of fast rotating stars

Stars may be understood as self-gravitating masses of a compressible fluid whose radiative cooling is compensated by nuclear reactions or gravitational contraction. The understanding of their time evolution requires the use of detailed models that account for a complex microphysics including that of opacities, equation of state and nuclear reactions. The present stellar models are essentially one-dimensional, namely spherically symmetric. However, the interpretation of recent data like the surface abundances of elements or the distribution of internal rotation have reached the limits of validity of one-dimensional models because of their very simplified representation of large-scale fluid flows. In this article, we describe the ESTER code, which is the first code able to compute in a consistent way a two-dimensional model of a fast rotating star including its large-scale flows. Compared to classical 1D stellar evolution codes, many numerical innovations have been introduced to deal with this complex problem. First, the spectral discretization based on spherical harmonics and Chebyshev polynomials is used to represent the 2D axisymmetric fields. A nonlinear mapping maps the spheroidal star and allows a smooth spectral representation of the fields. The properties of Picard and Newton iterations for solving the nonlinear partial differential equations of the problem are discussed. It turns out that the Picard scheme is efficient on the computation of the simple polytropic stars, but Newton algorithm is unsurpassed when stellar models include complex microphysics. Finally, we discuss the numerical efficiency of our solver of Newton iterations. This linear solver combines the iterative Conjugate Gradient Squared algorithm together with an LU-factorization serving as a preconditioner of the Jacobian matrix.

Why do stars form in clusters? An analytic model for stellar correlation functions

Recently, we have shown that if the ISM is governed by super-sonic turbulent flows, the excursion-set formalism can be used to calculate the statistics of self-gravitating objects over a wide range of scales. On the largest self-gravitating scales ('first crossing'), these correspond to GMCs, and on the smallest non-fragmenting self-gravitating scales ('last crossing'), to protostellar cores. Here, we extend this formalism to rigorously calculate the auto and cross-correlation functions of cores (and by extension, young stars) as a function of spatial separation and mass, in analogy to the cosmological calculation of halo clustering. We show that this generically predicts that star formation is very strongly clustered on small scales: stars form in clusters, themselves inside GMCs. Outside the binary-star regime, the projected correlation function declines as a weak power-law, until a characteristic scale which corresponds to the characteristic mass scale of GMCs. On much larger scales the clustering declines such that star formation is not strongly biased on galactic scales, relative to the actual dense gas distribution. The precise correlation function shape depends on properties of the turbulent spectrum, but its qualitative behavior is quite general. The predictions agree well with observations of young star and core autocorrelation functions over ~4 dex in radius. Clustered star formation is a generic consequence of supersonic turbulence if most of the power in the velocity field, hence the contribution to density fluctuations, comes from large scales. The distribution of self-gravitating masses near the sonic length is then imprinted by fluctuations on larger scales. We similarly show that the fraction of stars formed in 'isolated' modes should be small (\lesssim10%).

Фигуры равновесия жидких самогравитирующих неоднородных масс

Фигуры равновесия жидких самогравитирующих неоднородных масс

Centrifugal force: an appreciation

The centrifugal force is used to increase the physical understanding of five examples taken from fluid dynamics, geophysics and the solar system, as well as four hypothetical orbital problems. Each example involves a balance of forces between the centrifugal force and one or two other forces, such as a pressure gradient and a component of the force of gravity. Among the examples chosen for examination are: the orbital motion of fluid particles in surface grav-ity waves, the boundary layer character of steady flow next to a curved rigid surface, the tornado, the rotating self-gravitating mass and the three-body problem.

SELF-GRAVITATING ASTROPHYSICAL MASS WITH SINGULAR CENTRAL DENSITY VIBRATING IN FUNDAMENTAL MODE

The fluid-dynamical model of a self-gravitating mass of viscous liquid with singular density at the center vibrating in fundamental mode is considered in juxtaposition with that for Kelvin fundamental mode in a homogeneous heavy mass of incompressible inviscid liquid. Particular attention is given to the difference between spectral formulas for the frequency and lifetime of f-mode in the singular and homogeneous models. The newly obtained results are discussed in the context of theoretical asteroseismology of pre-white dwarf stage of red giants and stellar cocoons — spherical gas-dust clouds with dense star-forming core at the center.

Resonance Bands and Binary‐Star Formation

Numerical computations on the evolution of realistically stratified, asymmetric, self-gravitating masses reveal the onset of an instability that is suggestive of, but not decisive for, the formation of a binary star. On the other hand, analysis of the evolution and stability of idealized models consisting of uniform-density ellipsoids has been qualitatively accurate in predicting certain features of the behavior of the more realistically stratified figures. This idealized theory is, therefore, reconsidered in an attempt to isolate additional qualitative features that may be common both to the idealized and to realistic figures, and therefore, serve not only as a guide in formulating issues to be addressed in numerical computations but also as a means of interpreting the outcomes of such calculations. Several such features are isolated in this paper, the most striking of which is the existence and importance of resonance bands of instability encountered by the evolutionary trajectories.

Variability in the stellar initial mass function at low and high mass: three-component IMF models

Three-component models of the initial mass function (IMF) are made to consider possible origins for the observed relative variations in the numbers of brown dwarfs, solar-to-intermediate-mass stars and high-mass stars. The differences between the IMFs observed for clusters, field and remote field are also discussed. Three distinct physical processes that should dominate the three stellar mass regimes are noted. The characteristic mass for most star formation is identified with the thermal Jeans mass in the molecular cloud core, and this presumably leads to the middle mass range by the usual collapse and accretion processes. Pre-stellar condensations (PSCs) observed in millimetre-wave continuum studies presumably form at this mass. Significantly smaller self-gravitating masses require much larger pressures and may arise following dynamical processes inside these PSCs, including disc formation, tight-cluster ejection, and photoevaporation as studied elsewhere, but also gravitational collapse of shocked gas in colliding PSCs. Significantly larger stellar masses form in relatively low abundance by normal cloud processes, possibly leading to steep IMFs in low-pressure field regions, but this mass range can be significantly extended in high-pressure cloud cores by gravitationally focused gas accretion on to PSCs and by the coalescence of PSCs. These models suggest that the observed variations in brown dwarf, solar-to-intermediate-mass and high-mass populations are the result of dynamical effects that depend on environmental density and velocity dispersion. They accommodate observations ranging from shallow IMFs in cluster cores to Salpeter IMFs in average clusters and whole galaxies to steep and even steeper IMFs in field and remote field regions. They also suggest how the top-heavy IMFs in some starburst clusters may originate and they explain bottom-heavy IMFs in low surface brightness galaxies.

Helical fields and filamentary molecular clouds -- II. Axisymmetric stability and fragmentation

ABSTRA C T In Paper I (this issue), we constructed models of filamentary molecular clouds that are truncated by a realistic external pressure and contain a rather general helical magnetic field. We address the stability of our models to gravitational fragmentation and axisymmetric magnetohydrodynamically driven instabilities. By calculating the dominant modes of axisymmetric instability, we determine the dominant length-scales and growth rates for fragmentation. We find that the role of pressure truncation is to decrease the growth rate of gravitational instabilities by decreasing the self-gravitating mass per unit length. Purely poloidal and toroidal fields also help to stabilize filamentary clouds against fragmentation. The overall effect of helical fields is to stabilize gravity-driven modes, so that the growth rates are significantly reduced below what is expected for unmagnetized clouds. However, magnetohydrodynamic (MHD) ‘sausage’ instabilities are triggered in models in which the toroidal flux to mass ratio exceeds the poloidal flux to mass ratio by more than a factor of ,2. We find that observed filaments appear to lie in a physical regime where the growth rates of both gravitational fragmentation and axisymmetric MHD-driven modes are at a minimum.

On the Solutions of Liouville Systems

for some prescribed M>0, together with the Dirichlet boundary condition u=0 on 0. The Liouville equation finds applications in various fields of Physics and Mathematics. To name a few, it represents the Newtonian potential of a cluster of self-gravitating mass distribution where a>0 [1, 6, 36, 37], the electric potential induced by charge carriers in the theory of electrolytes where a<0 [31], and maximal entropy solutions of the incompressible Euler equation [23, 9]. In the field of Differential Geometry it stands for the problem of finding a metric corresponding to a prescribed Gaussian curvature [3, 10]. The system-analog of (L) takes the form

Analysis of nonradial pulsations of a homogeneous neutron star

Nonradial pulsations of an isolated neutron star were studied in two continuum models — a self-gravitating spherical mass of an inviscid, incompressible fluid and an elastic Fermi continuum. A detailed analytical derivation of natural spheroidal and torsional oscillation frequencies of a neutron star of mass 1.4MΦ based on notions about neutron matter as an elastic Fermi continuum is given. A comparison of numerical estimates of natural frequencies shows that both models predict the same order of pulsations ∼ 104 Hz; however, the frequencies of spheroidal modes calculated in the elastic Fermi continuum model are about 2–2.5 times higher than the frequencies calculated in the framework of the Kelvin hydrodynamic theory.

Dynamical stability of differentially rotating masses - III. Additional numerical results and interpretation

We investigate the dynamical stability of differentially rotating, self-gravitating masses. We obtain two types of shear instability. The first is caused by the ‘principal’ mode which is characterized by a peak growth rate and large azimuthal wavenumbers. This mode disappears for large shear or large flattening. We show that this instability can be described in terms of two parameters. The m = 2 principal mode achieves a non-linear evolution from an initially axisymmetric state towards a uniformly rotating Jacobi ellipsoid. The second type of instability is related to the ‘secondary’ modes. They form a discrete spectrum of corotating modes. They have a smaller growth rate than the principal mode for low values of m. Larger growth rates are obtained for large m. In the case of a flattened disc, we show that these modes can be identified with the unstable acoustic waves which are known to occur in compressible shear flows.

Non-axisymmetric dynamical instability of differentially rotating, self-gravitating masses

Summary We investigate the dynamical stability of differentially rotating, self-gravitating, homogeneous masses. Besides the classical bar-like instability we obtain a new non-axisymmetric Kelvin-Helmholtz-like instability which resembles the Papaloi-zou-Pringle instability in shearing accretion tori. We define the flattening parameter, f, as the ratio of the polar to the equatorial radius. The growth-rate curve can be divided into two components. For low angular velocities (f ≤ 1), the instability grows on a linear time-scale. The growth rate then rises towards a peak value. This occurs near the point where the uniformly rotating Maclaurin sequence exhibits a non-axisymmetric bifurcation point. For m = 2 this is the Jacobi bifurcation point. The instability is reduced for small values of f due to the stabilizing effect of the gravitational force. The results are almost independent of the form of the rotation law. Maximum instability occurs for intermediate values of m. A comparison is made with results obtained in the absence of self-gravitation. We discuss possible implications on stellar evolution

Approximate solution of third-order Clairaut theory of rotational sectorial figures of Jupiter and Saturn

The aim of this paper is to investigate numerical solutions of third-order Clairaut theory, under the boundary conditions given in our previous work (El-Shaarawy, 1974). This solution gives an explicit form of the shape and rotational distortion, due to third-order sectorial harmonic terms, of the equipotential surfaces of the two rapidly rotating planets, Jupiter and Saturn at the different levels inside these planets owing to a certain internal density distribution model (Zharkov, 1975). We considered each of them as a heterogeneous self-gravitating fluid mass in hydrostatic equilibrium.

Asymptotic properties of spherically symmetric self-gravitating mass systems for t → ∞

Abstract We prove some asymptotic results for the solutions of the Vlasov-Poisson system in the case of spherical symmetry as t→∞.

Self-similar flow behind a spherical shock with varying strength in an inhomogeneous self-gravitating medium

The self-similar model of a shock wave, produced on account of an instantaneous release of energy in an inhomogeneous self-gravitating gaseous mass, has been discussed with the help of equations of motion and equilibrium conditions. The disturbances are headed by a shock of variable strength. The variation of velocity density, pressure, and mass have been discussed for the different values of strength of the shock.

On the equilibrium of heated self-gravitating masses - Cooling by conduction

An investigation is given of the equilibrium states available to a self-gravitating mass of gas, cooling by conduction, and being heated at a rate proportional to the local gas density. The plane geometry situation is shown to be reducible to quadratures for the pressure, density, temperature, and gravitational potential. For a constant thermal conductivity it is shown that the gas density has either a central maximum or a central minimum, depending on the ratio of the thermal conductivity to a parameter taken to be a measure of the rate of heating. For a thermal conductivity which is a positive power of the temperature, it is shown that the gas density always has a central minimum and a maximum at the outer boundary of the configuration. For cylindrical and spherical geometrical configurations the same general properties are obtained. The physical origin of this behavior is discussed, and it is suggested that these exploratory calculations provide an effect which may not only aid in understanding thin filamentary structure observed in supernova remnants, but also help to assuage the difficulties of producing maser activity in the interior regions of 'cocoon' protostars.

On the equilibrium structures of self-gravitating masses of gas containing axisymmetric magnetic fields

The general equations describing the equilibrium shapes of self-gravitating gas clouds containing axisymmetric magnetic fields are presented. The general equations admit of a large class of solutions. It is shown that if one additional (ad hoc) asumption is made that the mass be spherically symmetrically distributed, then the gas pressure and the boundary conditions are sufficiently constraining that the general topological structure of the solution is effectively determined. The further assumption of isothermal conditions for this case demands that all solutions possess force-free axisymmetric magnetic fields. It is also shown how the construction of aspherical (but axisymmetric) configurations can be achieved in some special cases, and it is demonstrated that the detailed form of the possible equilibrium shapes depends upon the arbitrary choice of the functional form of the variation of the gas pressure along the field lines.

Rotating, Self-Gravitating Masses

Rotating, Self-Gravitating Masses

The growth of correlations in an expanding universe and the clustering of galaxies

From the BBGKY hierarchy of kinetic equations, an equation is derived for the growth of pair correlations in an expanding universe of self-gravitating point masses which is correlation-free at some initial time. This equation is solved explicitly and, from the full pair correlation function, the spatial correlation function is calculated. If t/sub 0/ is the time when clustering begins, the solutions are accurate until about 2t/sub 0/ when the mathematical approximations used for simplifying the kinetic equations no longer hold. During this period, the correlation function grows significantly. At time t, it has a characteristic length which is approximately (t-t/sub 0/) u, where u is the thermal speed of the galaxies. Although the correlation function derived from this analysis cannot be compared directly with the observed covariance function, its growth rate can be extrapolated. The results suggest that the present degree of galaxy clustering could have arisen from an initially unclustered state at the time of galaxy formation. (AIP)

Formation of Galaxies and Clusters of Galaxies by Self-Similar Gravitational Condensation

view Abstract Citations (4359) References (31) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Formation of Galaxies and Clusters of Galaxies by Self-Similar Gravitational Condensation Press, William H. ; Schechter, Paul Abstract We consider an expanding Friedmann cosmology containing a "gas" of self-gravitating masses. The masses condense into aggregates which (when sufficiently bound) we identify as single particles of a larger mass. We propose that after this process has proceeded through several scales, the mass spectrum of condensations becomes "self-similar" and independent of the spectrum initially assumed. Some details of the self-similar distribution, and its evolution in time, can be calculated with the linear perturbation theory. Unlike other authors, we make no ad hoc assumptions about the spectrum of long-wavelength initial perturbatidns: the nonlinear N-body interactions of the mass points randomize their positions and generate a perturbation to all larger scales; this should fix the self-similar distribution almost uniquely. The results of numerical experiments on 1000 bodies are presented; these appear to show new nonlinear effects: condensations can "bootstrap" their way up in size faster than the linear theory predicts. Our self-similar model predicts relations between the masses and radii of galaxies and clusters of galaxies, as well as their mass spectra. We compare the predictions with available data, and find some rather striking agreements. If the model is to explain galaxies, then isothermal "seed" masses of 3 x 1 0 M0 must have existed at recombination. To explain clusters of galaxies, the only necessary seeds are the galaxies themselves. The size of clusters determines, in principle, the deceleration parameter q0 presently available data give only very broad limits, unfortunately. Subject headings: cosmology - galaxies - galaxies, clusters of Publication: The Astrophysical Journal Pub Date: February 1974 DOI: 10.1086/152650 Bibcode: 1974ApJ...187..425P full text sources ADS | data products SIMBAD (6)