Self-similar dynamics of a magnetized polytropic gas

In broad astrophysical contexts of large-scale gravitational collapses and outflows and as a basis for various further astrophysical applications, we formulate and investigate a theoretical problem of self-similar magnetohydrodynamics (MHD) for a non-rotating polytropic gas of quasi-spherical symmetry permeated by a completely random magnetic field. Within this framework, we derive two coupled nonlinear MHD ordinary differential equations (ODEs), examine properties of the magnetosonic critical curve, obtain various asymptotic and global semi-complete similarity MHD solutions, and qualify the applicability of our results. Unique to a magnetized gas cloud, a novel asymptotic MHD solution for a collapsing core is established. Physically, the similarity MHD inflow towards the central dense core proceeds in characteristic manners before the gas material eventually encounters a strong radiating MHD shock upon impact onto the central compact object. Sufficiently far away from the central core region enshrouded by such an MHD shock, we derive regular asymptotic behaviours. We study asymptotic solution behaviours in the vicinity of the magnetosonic critical curve and determine smooth MHD eigensolutions across this curve. Numerically, we construct global semi-complete similarity MHD solutions that cross the magnetosonic critical curve zero, one, and two times. For comparison, counterpart solutions in the case of an isothermal unmagnetized and magnetized gas flows are demonstrated in the present MHD framework at nearly isothermal and weakly magnetized conditions. For a polytropic index γ=1.25 or a strong magnetic field, different solution behaviours emerge. With a strong magnetic field, there exist semi-complete similarity solutions crossing the magnetosonic critical curve only once, and the MHD counterpart of expansion-wave collapse solution disappears. Also in the polytropic case of γ=1.25, we no longer observe the trend in the speed-density phase diagram of finding infinitely many matches to establish global MHD solutions that cross the magnetosonic critical curve twice.      

Dynamic evolution of a quasi-spherical general polytropic magnetofluid with self-gravity

In various astrophysical contexts, we analyze self-similar behaviours of magnetohydrodynamic (MHD) evolution of a quasi-spherical polytropic magnetized gas under self-gravity with the specific entropy conserved along streamlines. In particular, this MHD model analysis frees the scaling parameter n in the conventional polytropic self-similar transformation from the constraint of n+γ=2 with γ being the polytropic index and therefore substantially generalizes earlier analysis results on polytropic gas dynamics that has a constant specific entropy everywhere in space at all time. On the basis of the self-similar nonlinear MHD ordinary differential equations, we examine behaviours of the magnetosonic critical curves, the MHD shock conditions, and various asymptotic solutions. We then construct global semi-complete self-similar MHD solutions using a combination of analytical and numerical means and indicate plausible astrophysical applications of these magnetized flow solutions with or without MHD shocks.     

Gravitational collapse of polytropic, magnetized, filamentary clouds

For the case in which the gas of a magnetized filamentary cloud obeys a polytropic equation of state, gravitational collapse of the cloud is studied using a simplified model. We concentrate on the radial distribution and restrict ourselves to a purely toroidal magnetic field. If the axial motions and poloidal magnetic fields are sufficiently weak, we could reasonably expect our solutions to be a good approximation. We show that while the filament experiences gravitational condensation and the density at the centre increases, the toroidal flux-to-mass ratio remains constant. A series of spatial profiles of density, velocity and magnetic field for several values of the toroidal flux-to-mass ratio and the polytropic index, is obtained numerically and discussed.     

New self-similar solutions of polytropic gas dynamics

We explore semicomplete self-similar solutions for the polytropic gas dynamics involving self-gravity under spherical symmetry, examine behaviours of the sonic critical curve and present new asymptotic collapse solutions that describe 'quasi-static' asymptotic behaviours at small radii and large times. These new 'quasi-static' solutions with divergent mass density approaching the core can have self-similar oscillations. Earlier known solutions are summarized. Various semicomplete self-similar solutions involving such novel asymptotic solutions are constructed, either with or without a shock. In contexts of stellar core collapse and supernova explosion, a hydrodynamic model of a rebound shock initiated around the stellar degenerate core of a massive progenitor star is presented. With this dynamic model framework, we attempt to relate progenitor stars and the corresponding remnant compact stars: neutron stars, black holes and white dwarfs.     

On X-ray variability in narrow-line and broad-line active galactic nuclei

We propose a novel mathematical method to construct an exact polytropic sphere in self-gravitating hydrostatic equilibrium, improving the non-linear Poisson equation. The central boundary condition for the present equation requires a ratio of gas pressure to total one at the centre, which is uniquely identified by the whole mass and molecular weight of the system. The special solution derived from the Lane–Emden equation can be reproduced. This scheme is now available for modelling the molecular cloud cores in interstellar media. The mass–radius relation of the first core is found to be consistent with the recent results of radiation hydrodynamic simulations.     

Structures in a class of magnetized scale-free discs

We construct analytically stationary global configurations for both aligned and logarithmic spiral coplanar magnetohydrodynamics (MHD) perturbations in an axisymmetric background MHD disc with a power-law surface mass density  Σ₀∝ r ^−α  , a coplanar azimuthal magnetic field   B ₀∝ r ^−γ  , a consistent self-gravity and a power-law rotation curve   v ₀∝ r ^−β  , where v ₀ is the linear azimuthal gas rotation speed. The barotropic equation of state  Π∝Σ ⁿ   is adopted for both MHD background equilibrium and coplanar MHD perturbations where Π is the vertically integrated pressure and n is the barotropic index. For a scale-free background MHD equilibrium, a relation exists among  α, β, γ  and n such that only one parameter (e.g. β) is independent. For a linear axisymmetric stability analysis, we provide global criteria in various parameter regimes. For non-axisymmetric aligned and logarithmic spiral cases, two branches of perturbation modes (i.e. fast and slow MHD density waves) can be derived once β is specified. To complement the magnetized singular isothermal disc analysis of Lou, we extend the analysis to a wider range of  −1/4 < β < 1/2  . As an illustrative example, we discuss specifically the  β= 1/4  case when the background magnetic field is force-free. Angular momentum conservation for coplanar MHD perturbations and other relevant aspects of our approach are discussed.     

General relativistic self-similar solutions in cosmology

We present general relativistic solutions for self-similar spherical perturbations in an expanding cosmological background of cold pressure-less gas. We focus on solutions having shock discontinuities propagating in the surrounding cold gas. The pressure, p , and energy density, μ, in the shock-heated matter are assumed to obey   p = w μ  , where w is a positive constant. Consistent solutions are found for shocks propagating from the symmetry centre of a region of a positive density excess over the background. In these solutions, shocks exist outside the radius marking the event horizon of the black hole which would be present in a shock-less collapse. For large jumps in the energy density at the shock, a black hole is avoided altogether and the solutions are regular at the centre. The shock-heated gas does not contain any sonic points, provided the motion of the cold gas ahead of the shock deviates significantly from the Hubble flow. For shocks propagating in the uniform background, sonic points always appear for small jumps in the energy density. We also discuss self-similar solutions without shocks in fluids with   w < −1/3  .     

Numerical modeling of coronal mass ejections based on various pre-event model atmospheres

We examine how the initial state (pre-event corona) affects the numerical MHD simulation for a coronal mass ejection (CME). Earlier simulations based on a pre-event corona with a homogeneous density and temperature distribution at the lower boundary (i.e., solar surface) have been used to analyze the role of streamer properties in determining the characteristics of loop-like transients. The present paper extends these studies to show how a broader class of global coronal properties leads not only to different types of CMEs, but also modifies the adjacent quiet corona and/or coronal holes.We consider four pre-event coronal cases: (1) constant boundary conditions and a polytropic gas with = 1.05; (2) non-constant (latitude dependent) boundary conditions and a polytropic gas with = 1.05; (3) constant boundary conditions with a volumetric energy source and = 1.67; (4) non-constant (latitude dependent) boundary conditions with a volumetric energy source and = 1.67. In all models, the pre-event magnetic fields separate the corona into closed field regions (streamers) and open field regions. The CME's initiation is simulated by introducing at the base of the corona, within the streamer region, a standard pressure pulse and velocity change. Boundary values are determined using MHD characteristic theory.The simulations show how different CMEs, including loop-like transients, clouds and bright rays, might occur. There are significant new features in comparison to published results. We conclude that the pre-event corona is a crucial factor in dictating CMEs properties.     

Scalar perturbations in two-temperature cosmological plasmas

We study the properties of density perturbations of a two-component plasma with a temperature difference on a homogeneous and isotropic background. For this purpose, we extend the general relativistic gauge-invariant and covariant (GIC) perturbation theory to include a multifluid with a particular equation of state (ideal gas) and imperfect fluid terms due to the relative energy flux between the two species. We derive closed sets of GIC vector and subsequently scalar evolution equations. We then investigate solutions in different regimes of interest. In particular, we study long-wavelength and arbitrary-wavelength Langmuir and ion-acoustic perturbations. The harmonic oscillations are superposed on a Jeans-type instability. We find a generalized Jeans criterion for collapse in a two-temperature plasma, which states that the species with the largest sound velocity determines the Jeans wavelength. Furthermore, we find that within the limit for gravitational collapse, initial perturbations in either the total density or charge density lead to a growth in the initial temperature difference. These results are relevant for the basic understanding of the evolution of inhomogeneities in cosmological models.     

Mass-Temperature Relation of X-ray Clusters in Triaxial Halo

We investigate the mass-temperature relation of clusters for both the spherical NFW halo model and a concentric triaxial halo model. We study the temperature and density distributions of both an isothermal and a polytropic intra-cluster gas in hydrostatic equilibrium. We find that both the uncertainties in the concentration parameter and in the eccentricities (in case of the triaxial halo) lead to a greater scatter in the emission-weighted temperature at a given halo mass for less massive clusters. This may be helpful when determining the intrinsic statistical error of the σ8 normalization of the linear power spectrum from cluster abundance.     

The structure of MHD shock waves in a thermally-radiating gas at high mach numbers

The structure of a strong MHD shock wave which radiates thermally downstream of the shock is studied by asymptotic expansion. The exact integral equation for radiation is adopted for the study. Hence, the optically thick (and thin), the general differential approximate and the exact integral equation solutions may now be compared.     

General polytropic spheres as gravitational lenses

We propose hydrostatic polytropic spheres governed by the Lane-Emden equation (LEE) of index $n$ as a novel set of physical models for axially averaged gravitational lenses anywhere in the Universe, alternative to the familiar singular isothermal sphere (SIS) and the Navarro–Frenk–White (NFW) profile, as such general polytropic spheres are conceptually simple, versatile in representing a series of equations of state, and able to address both the inner core and cusp features. As LEE is nonlinear, there exist several distinct classes of LEE solutions to serve as physical lens models. With a few scaling parameters, the complete problem can be readily reconstructed with full physical dimensions. A given mass density profile satisfying LEE produces lensing effects that are solely determined by a dimensionless parameter $q$ which contains geometric and kinematic information about the source-lens-observer system. The lens mapping and tangential shear or distortion profile are derived, first analytically for special cases and then asymptotically at the outskirts or near the edge of the lens. Numerical procedures for calculating full lensing profiles of a general lens are developed. Our results include the analytical “singular polytropic sphere” (SPS) profile which generalizes the SIS model and may outperform the latter in modeling dark matter halos among others. We further point out that dynamic models of general polytropic spheres in self-similar evolution can serve as several broad classes of gravitational lenses and produce time-dependent lensing effects slow or fast depending on the pertinent time scales. Astrophysical sources that can be lensed include electromagnetic wave sources in the entire frequency band, gravitational wave sources in the entire frequency band, gravitons even possibly with finite masses, neutrino sources of three different types, neutron sources, and ultra high energy cosmic rays (UHECRs) of electrically charged particles which can also interact with magnetic fields. We discuss and elabrate applications to dark matter halos, hypermassive black holes and supermassive black holes in the entire Universe including the early Universe, magnetized supermassive stars, static and dynamically evolving spherical and cylindrical lenses in contexts of astrophysics and cosmology.     

X-ray clusters of galaxies in conformal gravity

We run adiabatic N -body/hydrodynamical simulations of isolated self-gravitating gas clouds to test whether conformal gravity, an alternative theory to general relativity, is able to explain the properties of X-ray galaxy clusters without resorting to dark matter. We show that the gas clouds rapidly reach equilibrium with a density profile which is well fitted by a β-model whose normalization and slope are in approximate agreement with observations. However, conformal gravity fails to yield the observed thermal properties of the gas cloud: (i) the mean temperature is at least an order of magnitude larger than the observed and (ii) the temperature profiles increase with the square of the distance from the cluster centre, in clear disagreement with real X-ray clusters. These results depend on a gravitational potential whose parameters reproduce the velocity rotation curves of spiral galaxies. However, this parametrization stands on an arbitrarily chosen conformal factor. It remains to be seen whether a different conformal factor, specified by a spontaneous breaking of the conformal symmetry, can reconcile this theory with observations.     

On the r-mode spectrum of relativistic stars: the inclusion of the radiation reaction

We consider both mode calculations and time-evolutions of axial r modes for relativistic uniformly rotating non-barotropic neutron stars, using the slow-rotation formalism, in which rotational corrections are considered up to linear order in the angular velocity Ω. We study various stellar models, such as uniform density models, polytropic models with different polytropic indices n , and some models based on realistic equations of state. For weakly relativistic uniform density models and polytropes with small values of n , we can recover the growth times predicted from Newtonian theory when standard multipole formulae for the gravitational radiation are used. However, for more compact models, we find that relativistic linear perturbation theory predicts a weakening of the instability compared to the Newtonian results. When turning to polytropic equations of state, we find that for certain ranges of the polytropic index n , the r mode disappears, and instead of a growth, the time-evolutions show a rapid decay of the amplitude. This is clearly at variance with the Newtonian predictions. It is, however, fully consistent with our previous results obtained in the low-frequency approximation.     

BeppoSAX–ROSAT PSPC observations of the Shapley supercluster: A3562

We present a combined analysis of the BeppoSAX and ROSAT PSPC observation of the cluster of galaxies A3562, a massive member of the core of the Shapley supercluster. With a complex and interacting structure composed from two groups of galaxies and A3558 to the west, the surface brightness of A3562 shows excess in the sectors to east and south when compared with an azimuthally averaged model of the emission. The emission tends to be flatter, and the distribution of the gas broader, along the merging axis and in opposition to the two groups. We present the first determination of the gradients of the gas temperature and metallicity for a cluster in the Shapley region at large distance from the cluster centre. From an analysis of the BeppoSAX data in annuli and sectors, we observe both the profiles to be flat within 8 arcmin (∼0.62 Mpc), with emission-weighted values of kT =5.1±0.2 keV and Z =0.39±0.05 Z_⊙. The value of the temperature is consistent with recent ASCA measurements and is significantly higher than previous estimates obtained from ROSAT and EXOSAT . We discuss the possible reasons for this disagreement. Between 8 and 20 arcmin, the plasma temperature declines to about 3.2 keV. When a polytropic profile is used to represent the gas temperature profile, the best-fitting polytropic index is 1.16±0.03. These results imply a total mass within the virial radius of between 40 and 80 per cent lower than the optical estimate, and a gas mass fraction of about 30 per cent.     

Compressible magnetohydrodynamic turbulence: mode coupling, scaling relations, anisotropy, viscosity-damped regime and astrophysical implications

We present numerical simulations and explore scalings and anisotropy of compressible magnetohydrodynamic (MHD) turbulence. Our study covers both gas-pressure-dominated (high β) and magnetic-pressure-dominated (low β) plasmas at different Mach numbers. In addition, we present results for super-Alfvénic turbulence and discuss in what way it is similar to sub-Alfvénic turbulence. We describe a technique of separating different magnetohydrodynamic modes (slow, fast and Alfvén) and apply it to our simulations. We show that, for both high- and low-β cases, Alfvén and slow modes reveal a Kolmogorov   k ^−5/3  spectrum and scale-dependent Goldreich–Sridhar anisotropy, while fast modes exhibit a   k ^−3/2  spectrum and isotropy. We discuss the statistics of density fluctuations arising from MHD turbulence in different regimes. Our findings entail numerous astrophysical implications ranging from cosmic ray propagation to gamma ray bursts and star formation. In particular, we show that the rapid decay of turbulence reported by earlier researchers is not related to compressibility and mode coupling in MHD turbulence. In addition, we show that magnetic field enhancements and density enhancements are marginally correlated. Addressing the density structure of partially ionized interstellar gas on astronomical-unit scales, we show that the viscosity-damped regime of MHD turbulence that we reported earlier for incompressible flows persists for compressible turbulence and therefore may provide an explanation for these mysterious structures.     

Studies of accretion flows around rotating black holes – II. Standing shocks in the pseudo-Kerr geometry

Standing, propagating or oscillating shock waves are common in accretion and winds around compact objects. We study the topology of all possible solutions using the pseudo-Kerr geometry. We present the parameter space spanned by the specific energy and angular momentum and compare it with that obtained from the full general relativity to show that the potential can work satisfactorily in fluid dynamics also, provided the polytropic index is suitably modified. We then divide the parameter space depending on the nature of the solution topology. We specifically study the nature of the standing Rankine–Hugoniot shocks. We also show that as the Kerr parameter is increased, the shock location generally moves closer to the black hole. In future, these solutions can be used as guidelines to test numerical simulations around compact objects.