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For filamentary clouds on various scales obeying general polytropic (GP) equation of state, their hydrodynamic collapses, expansions, and shocks are investigated. Our cylindrical model is axisymmetric, infinitely long with axial uniformity and involves Newtonian gravity. For such GP cylinders, we explore various analytical and numerical similarity solutions. Based on a singular hydrostatic solution, we derive a quasi-static asymptotic dynamic solution approaching the axis. There, we also derive the asymptotic cylindrical free-fall solution for polytropic index γ  ≤ 1 and show the absence of such solutions for γ > 1. We find new asymptotic solutions for expanding cylindrical central voids with no matter inside, and examine the asymptotic expansion solutions to higher orders far from the axis. We classify the sonic critical curve (SCC) into three (or five) types and analyse their properties. The asymptotic behaviors of the SCC towards the axis and infinity are examined. Examples are shown for solutions crossing the SCC twice with the global features of cylindrical envelope expansion or contraction with core collapses. We numerically construct new types of global similarity solutions with or without outgoing shocks. For γ > 1, a shock is necessary to connect the inner and outer parts. The collapse and fragmentation of massive filaments or strings may give clues and implications to the formations of chains of stellar objects, chains of black holes, chains of galaxies or even chains of galaxy clusters in proper astrophysical and cosmological contexts. [ABSTRACT FROM AUTHOR]

We investigate and explore self-similar dynamic radial collapses of relativistic degenerate stellar cores (RDSCs) and radiation pressure dominated stellar interiors (RPDSIs) of spherical symmetry by invoking a conventional polytropic (CP) equation of state (EoS) with a constant polytropic index γ = 4 / 3 and by allowing free-fall non-zero RDSC or RPDSI surface mass density and pressure due to their sustained physical contact with the outer surrounding stellar envelopes also in contraction. Irrespective of the physical triggering mechanisms (including, e.g., photodissociation, electron-positron pair instability, general relativistic instability etc.) for initiating such a self-similar dynamically collapsing RDSC or RPDSI embedded within a massive star, a very massive star (VMS) or a supermassive star (SMS) in contraction and by comparing with the Schwarzschild radii associated with their corresponding RDSC/RPDSI masses, the emergence of central black holes in a wide mass range appears inevitable during such RDSC/RPDSI dynamic collapses inside massive stars, VMSs, and SMSs, respectively. Radial pulsations of progenitor cores or during a stellar core collapse may well leave imprints onto collapsing RDSCs/RPDSIs towards their self-similar dynamic evolution. Massive neutron stars may form during dynamic collapses of RDSC inside massive stars in contraction under proper conditions. [ABSTRACT FROM AUTHOR]

In several astrophysical and cosmological contexts, long dynamic filamentary structures emerge ubiquitously in observations and numerical simulations. For theoretical understanding, such filaments may be idealized as self-similar hydrodynamic cylinders under self-gravity of infinite length with axisymmetry and axial uniformity. We review and present analytical results of such a self-similar dynamic cylinder, including asymptotic solutions both towards the symmetry axis and at large radii, asymptotic solution just outside a dynamically expanding central void cylinder and a dimensional energy conservation equation. We introduce a velocity potential and derive cylindrical Bernoulli relations in similarity forms. Besides, the sonic critical curve (SCC) for γ = 1 is treated and examined. A thorough classification of the SCC of this kind is done. Most importantly, intrinsic invariance of the general polytropic dynamic cylinder emerges when γ = 1 , which builds connections among individual self-similar solutions of the same kind. We re-examine several key aspects of the study by Holden et al. (2009) and find that some of their results and conclusions to be misleading and incomplete. We pinpoint the sources of their errors and offer our correct results. Finally, we numerically compute self-similar solutions with a central free-fall behaviour as well as solutions that go across either the upper or the lower branch of the SCC once and show the corresponding global solution profiles. • We review and present analytical results of a self-similar dynamic cylinder under self-gravity, including asymptotic solutions both towards the symmetry axis and at large radii, asymptotic solution just outside a dynamically expanding central void cylinder and the Bernoulli relation. • We re-examine several key aspects of the model study by Holden et al. (2009) and find that some of their results and conclusions to be misleading and incomplete. • We pinpoint the sources of their errors and offer our correct results. • We numerically compute self-similar solutions with a central free-fall behaviour as well as solutions that go across either the upper or the lower branch of the sonic critical curve once and show the corresponding global solution profiles. [ABSTRACT FROM AUTHOR]

We explore novel self-similar dynamic evolution of converging spherical shocks in a self-gravitating isothermal gas under conceivable astrophysical situations. The construction of such converging shocks involves a time-reversal operation on feasible flow profiles in self-similar expansion with a proper care for the increasing direction of the specific entropy. Pioneered by Guderley since 1942 but without self-gravity so far, self-similar converging shocks are important for implosion processes in aerodynamics, combustion, and inertial fusion. Self-gravity necessarily plays a key role for grossly spherical structures in very broad contexts of astrophysics and cosmology, such as planets, stars, molecular clouds (cores), compact objects, planetary nebulae, supernovae, gamma-ray bursts, supernova remnants, globular clusters, galactic bulges, elliptical galaxies, clusters of galaxies as well as relatively hollow cavity or bubble structures on diverse spatial and temporal scales. Large-scale dynamic flows associated with such quasi-spherical systems (including collapses, accretions, fall-backs, winds and outflows, explosions, etc.) in their initiation, formation, and evolution are likely encounter converging spherical shocks at times. Our formalism lays an important theoretical basis for pertinent astrophysical and cosmological applications of various converging shock solutions and for developing and calibrating numerical codes. As examples, we describe converging shock triggered star formation, supernova explosions, and void collapses. [ABSTRACT FROM AUTHOR]

We study the hydrodynamic self-similar mass collapses of general polytropic (GP) spherical clouds to central Schwarzschild black holes and void evolution with or without shocks. In order to grossly capture characteristic effects of general relativity outside yet close to the event horizon of a Schwarzschild black hole and to avoid mathematical complexity, we adopt the approximation of the Paczynski–Wiita gravity to replace the simple Newtonian gravity in our model formulation. A new dimensionless parameter s appears with the physical meaning of the square of the ratio of the sound speed to the speed of light c. Various self-similar dynamic solutions are constructed for a polytropic index γ > 4/3. Two (for small enough s < 1) or no (for large enough s < 1) expansion-wave collapse solutions with central event horizons exist when γ > 4/3, representing the collapse of static singular GP spheres towards the central singularity of space–time. Such GP spherical dynamic mass collapse is shown to be highly efficient for the rapid formation of supermassive black holes (mass range of ∼106–1010 M⊙) in the early Universe or even hypermassive black holes (mass range of ∼1010–1012 M⊙) if extremely massive mass reservoirs could be sustained for a sufficiently long time, which may evolve into hard X-ray/gamma-ray sources or quasars according to their surroundings. Self-similar dynamic solutions of a GP gas are also proposed for the stellar mass black hole formation during the violent supernova explosion of a massive progenitor star, the time-scale of which is estimated of ∼10−3 s. Rebound shocks travelling in supernovae are also discussed based on our self-similar shock expansion solutions. [ABSTRACT FROM AUTHOR]

ABSTRACT There have been reports of possible detections of intermediate-mass black holes (IMBHs) in globular clusters (GCs). Empirically, there exists a tight correlation between the central supermassive black hole (SMBH) mass and the mean velocity dispersion of elliptical galaxies, 'pseudo-bulges' and classical bulges of spiral galaxies. We explore such a possible correlation for IMBHs in spherical GCs. In our model of self-similar general polytropic quasi-static dynamic evolution of GCs, a heuristic criterion of forming an IMBH is proposed. The key result is , where MBH is the IMBH mass, σ is the GC mean stellar velocity dispersion, is a coefficient and 2/3 < n < 1. Available observations are examined. [ABSTRACT FROM AUTHOR]

ABSTRACT A spherically symmetric hydrodynamic stellar core collapse process under gravity is time-dependent and may become unstable once disturbed. Subsequent non-linear evolutions of such growth of hydrodynamic instabilities may lead to various physical consequences. Specifically for a homologous collapse of a stellar core characterized by a polytropic exponent Γ= 4/3, we examine oscillations and/or instabilities of three-dimensional (3D) general polytropic perturbations. Being incompressible, the radial component of vorticity perturbation always grows unstably during the same homologous core collapse. For compressible 3D perturbations, the polytropic index γ of perturbations can differ from Γ= 4/3 of the general polytropic hydrodynamic background flow, where the background specific entropy is conserved along streamlines and can vary in radius and time. Our model formulation here is more general than previous ones. The Brunt-Väisälä buoyancy frequency does not vanish, allowing for the existence of internal gravity g − modes and/or g + modes, depending on the sign of respectively. Eigenvalues and eigenfunctions of various oscillatory and unstable perturbation modes are computed, given asymptotic boundary conditions. As studied in several specialized cases of Goldreich & Weber and of Lou & Cao and Cao & Lou, we further confirm that acoustic p modes and surface f modes remain stable in the current more general situations. In comparison, g − modes and sufficiently high radial order g + modes are unstable, leading to inevitable convective motions within the collapsing stellar interior; meanwhile, sufficiently low radial order g + modes remain stably trapped in the collapsing core. Unstable growths of 3D g-mode disturbances are governed dominantly by the angular momentum conservation and modified by the gas pressure restoring force. We note in particular that unstable temporal growths of 3D vortical perturbations exist even when the specific entropy distribution becomes uniform and γ=Γ= 4/3. Conceptually, unstable g modes might bear conceivable physical consequences on supernova explosions, the initial kicks of nascent proto-neutron stars of as high as up to and breakups of the collapsing core, while unstable growths of vortical perturbations can lead to fast spins of compact objects, 3D vortical convections inside the collapsing core for possible magnetohydrodynamic dynamo actions on seed magnetic fields, and the generation of Rossby waves further stimulated by gravitational wave emissions. [ABSTRACT FROM AUTHOR]

ABSTRACT With proper physical mechanisms of energy and momentum input from around the centre of a self-gravitating polytropic gas sphere, a central spherical 'void' or 'cavity' or 'bubble' of very much less mass contents may emerge and then dynamically expand into a variety of surrounding more massive gas envelopes with or without shocks. We explore self-similar evolution of a self-gravitating polytropic hydrodynamic flow of spherical symmetry with such an expanding 'void' embedded around the centre. The void boundary supporting a massive envelope represents a pressure-balanced contact discontinuity where drastic changes in mass density and temperature occur. We obtain numerical void solutions that can cross the sonic critical surface either smoothly or by shocks. Using the conventional polytropic equation of state, we construct global void solutions with shocks travelling into various envelopes including static polytropic sphere, outflow, inflow, breeze and contraction types. In the context of supernovae, we discuss the possible scenario of separating a central collapsing compact object from an outgoing gas envelope with a powerful void in dynamic expansion. Initially, a central bubble is carved out by an extremely powerful neutrinosphere. After the escape of neutrinos during the decoupling, the strong electromagnetic radiation field and/or electron-positron pair plasma continue to drive the cavity expansion. In a self-similar dynamic evolution, the pressure across the contact discontinuity decreases with time to a negligible level for a sufficiently long lapse, and eventually the gas envelope continues to expand by inertia. We describe model cases of polytropic index γ= 4/3 −ε with ε > 0 and discuss pertinent requirements to justify our proposed scenario. [ABSTRACT FROM AUTHOR]

BSTRACT [ABSTRACT FROM AUTHOR]

Spectral observations of molecular line profiles reveal the so-called 'blue profiles' for double-peaked molecular spectral lines with stronger blue and weaker red peaks as notable features for star-forming cloud core collapses under the self-gravity. In contrast, per cent of observed molecular spectral line profiles in star-forming clouds or cores also show the so-called double-peaked 'red profiles' with red peaks stronger than blue peaks. Gao & Lou show that these unexplained 'red profiles' can be signatures of global self-similar dynamics for envelope expansion with core collapse (EECC) within star-forming molecular clouds or cores. We demonstrate here that spatially resolved 'red profiles' of HCO () and CS () molecular transitions from the low-mass star-forming cloud core FeSt 457 together with its radial profile of column density inferred from dust extinction observations appear to reveal a self-similar hydrodynamic shock phase for global EECC. Observed spectral profiles of CO () are also fitted by the same EECC model. For further observational tests, the spatially resolved profiles of molecular transitions HCO () and CS () as well as the radial profiles of (sub)millimetre continuum emissions at three wavelengths of 1.2, 0.85 and 0.45 mm from FeSt 457 are also predicted. [ABSTRACT FROM AUTHOR]

ABSTRACT We present the theoretical results of a piecewise isothermal shock wind model. This was devised to predict the luminosity and surface brightness profile of diffuse X-ray emissions, primarily from the inner shocked downstream wind zone of a planetary nebula (PN) surrounded by a self-similar shocked dense shell and self-similar outer slow asymptotic giant branch wind envelope, both involving self-gravity. We compare and fit our computational model results with the available observations of a few grossly spherical X-ray emitting PNe. By matching the shocked piecewise isothermal self-similar void solutions with the self-gravity of Lou and Zhai for the outer zone and a stationary isothermal fast tenuous wind with a reverse shock for the inner zone across an expanding contact discontinuity, we can consistently construct dynamic evolution models of PNe with diffuse X-ray emissions. On the basis of such a chosen dynamic wind interaction model, both the X-ray luminosity and the radial X-ray brightness profile are determined by three key parameters: the so-called X-ray parameter, X, and two radii, Rrs and Rc, of the reverse shock and the contact discontinuity. We find that the morphologies of X-ray emissions appear in the forms of either a central luminous sphere or a bright ring embedded within optically bright shells. In contrast to previous adiabatic models, the X-ray brightness peaks around the reverse shock, instead of the contact discontinuity surface just inside the outer shocked dense shell. The diffuse X-ray emissions of a few observed PNe appear to support this piecewise isothermal wind-wind dynamic interaction scenario with shocks. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]