A simple model of a supernova

In this paper we formulate the problem of the collapse of a spherically-symmetric, radiating body in general relativity. The requirement that the metric and its normal derivative be continuous across the boundary imposes conditions upon the evolution of the star and allows identification of physical phenomena measured by a distant observer. A solution to Einstein's field equations for the exterior of a spherically-symmetric radiating body is that derived originally by Vaidya in 1951. By requiring the continuity described above we identify the mass, luminosity, velocity, and time increment measured by a distant observer in terms of the metric parameters evaluated in a frame comoving with the outer boundary. We also assume that the interior metric is a sum of products of functions of the radius and time. The continuity requirements allow the evolution of two of the three functions of time to be determined. The evolution of the third function, describing the motion of the core, is determined by the imposition of an equation of state at the center. The adiabatic index derived from the Baym-Bethe-Pethick equation of state was used to provide this last equation. A major result is obtaining an analytic solution to Einstein's field equations describing the core of a collapsing star. As a consequence of this solution we found that for the relatively small values of the adiabatic index (_max1.6), the star smoothly made the transition to a final collapsed state. Neither bounce nor shock wave was obtained. Also, there is a readily understood connection between the adiabatic index, and such parameters of the edge of the core as the velocity and acceleration. Finally, the analytic solutions provide the time-scales for the collapse which are significantly different from that of free-fall. The retarding effects of pressure upon the collapse are apparent. It is hoped that such analytic solutions will provide insight into more complicated dynamic systems in general relativity.     

On the stability of a class of radiating viscous self-gravitating stars with axial symmetry

This study investigates the stability of a class of radiating viscous self-gravitating stars with axial symmetry having anisotropic pressure. We use perturbation technique to establish the perturbed form of the Einstein field equations and dynamical equations. The instability range in the Newtonian and post-Newtonian eras has been analyzed by constructing the collapse equation. It is found that the adiabatic index has a key role in the discussion of instability ranges which depends upon the physical parameters, i.e., energy density, anisotropic pressure and shear viscosity of the fluid and heat flux. We conclude that the shear viscosity decreases the instability range and makes the system more stable.     

Fragmentation of a Molecular Cloud Core versus Fragmentation of the Massive Protoplanetary Disk in the Main Accretion Phase

We present three-dimensional numerical simulations on binary formation through fragmentation. The simulations follow gravitational collapse of a molecular cloud core up to growth of the first core by accretion. At the initial stage, the gravity is only slightly dominant over the gas pressure. We made various models by changing initial velocity distribution (rotation speed, rotation law, and bar-mode perturbation). The cloud fragments whenever the cloud rotates sufficiently slowly to allow collapse but faster enough to form a disk before first-core formation. The latter condition is equivalent to Ω₀ t _ff ? 0.05, where Ω₀ and t _ff f denote the initial central angular velocity and the freefall time measured from the central density, and the condition is independent of the initial rotation law and bar-mode perturbation. Fragmentation is classified into six types. When the initial cloud rotates rigidly the cloud collapses to form a adiabatic disk supported by rotation. When the bar-mode perturbation is very minor, the disk deforms to a rotating bar, and the bar fragments. Otherwise, the adiabatic disk evolves into a central core surrounded by a circumstellar disk, and the the circumstellar disk fragments. When the initial cloud rotates differentially, the cloud deforms to a ring or bar in the isothermal collapse phase. The ring fragments into free or more cores, while the bar fragments into only two cores. In the latter case, the core merges due to low orbital angular momentum and new satellite cores form in the later stages.     

On the opaque core appearing in the early phase of star formation

The spherical collapse of a protostar with one solar mass is calculated from a gravitationally unstable initial stage, by solving the equations of hydrodynamics and pherical radiative transfer. The temperature dependence of the dust opacity is taken into account in contrast to the earlier calculations with temperature independent opacities. It is shown that the opaque core is equivalent to the adiabatic core in the purely adiabatic case. The blanketing effect of dust grains strengthens the shock of the opaque core and may result in raising the central entropy of protostars.Paper presented at the Conference on Protostars and Planets, held at the Planetary Science Institute, University of Arizona, Tucson, Arizona, between January 3 and 7, 1978.     

Effect of equation of state and shock energy loss on SN II explosions

We consider the effect on the adiabatic gravitational collapse of stellar cores of large mass due to (1) the use of different equations of state and (2) the inclusion of energy loss of shock. We find that different equations of state give very different central densities at rebounce and amplitudes of rebounce. When shock energy loss is not considered, all the equations discussed here will lead to explosions, but when it is considered, small amplitude rebounces will not do so.     

A generic dynamical model of gamma-ray burst remnants

The conventional generic model is considered to explain the dynamics of gamma-ray burst remnants very well, no matter whether they are adiabatic or highly radiative. However, we find that, for adiabatic expansion, the model cannot reproduce the Sedov solution in the non-relativistic phase, and thus it needs to be revised. In this paper a new differential equation is derived. The generic model based on this equation is shown to be correct for both radiative and adiabatic fireballs, and in both ultrarelativistic and non-relativistic phases.     

Analysis of Two-Parameter Families of Triple Close ApproachesOccurring in Stellar Systems (I)

The analysis of two-parameter families of triple close approaches occurring in stellar systems is studied in a series of three papers. This paper deals with the role of triple encounters with low initial velocities and equal masses in the evolution of stellar systems. It shows how a condition of complete collapse may be perturbed to obtain well-established two-parameter families of asymmetric triple close approaches with the formation of a binary and with systematic regularity of escape of the third body. Our results also indicate that the conjecture of Szebehely viz., `The measure of escaping orbits is significantly higher than the measure of stable orbits' is likely to be true. Further our results differ from that of Agekian's escape probability criterian. The second paper deals with the role of triple encounters with low initial velocities and equal masses in the evolution of stellar systems in 3D space. The third and last paper offers applications in stellar systems. This revised version was published online in July 2006 with corrections to the Cover Date.     

Radiating gravitational collapse with shear viscosity

A model is proposed of a collapsing radiating star consisting of an isotropic fluid with shear viscosity undergoing radial heat flow with outgoing radiation. The pressure of the star, at the beginning of the collapse, is isotropic but owing to the presence of the shear viscosity the pressure becomes more and more anisotropic. The behaviour of the density, pressure, mass, luminosity and the effective adiabatic index is analysed. Our work is compared to the case of a collapsing shearing fluid of a previous model, for a star with 6 M_⊙.     

Capture into resonance: An extension of the use of adiabatic invariants

The theory of the adiabatic invariant predicts the long term evolution of mechanical systems with slowly varying parameters.Unfortunately, it is not valid when the system goes across a critical trajectory. This case is important because it can lead to capture into resonance.Analysing the motion in the vincinity of the critical trajectory, we are able to give general formulae for the probability of capture and to show that in general, the adiabatic invariant is conserved (allowance being made for the geometrical discontinuity in its definition at the critical orbit).     

The phase transitions of superdense matter and supernova explosions

Effects of the phase transitions of superdense matter on supernova explosions are investigated with the aid of an idealized equation of state on the assumptions of adiabatic collapse. It is found that in the case of strong phase transitions explosions become weaker, while in the case of weak phase transitions explosions become stronger. However, the increment of the ejected energy is not so large as suggested by Migdalet al. (1979).Paper presented at the IAU Third Asian-Pacific Regional Meeting, held in Kyoto, Japan, between 30 September–6 October, 1984.     

On the Contraction of Protostellar Clouds with Different Metallicities

We examine the thermal and chemical evolution of gravitationally collapsing protostellar clouds with metallicity 0≤Z/Z _⊙≤1.During the first collapse stage, the temperatures are higher for lower metallicity clouds. However, in the course of the adiabatic contraction of transient cores, the evolutionary trajectories of the clouds converge to a curve that is determined only by fundamental physical constants. The trajectories coincide each other thereafter. The size of the stellar core at formation is the same regardless of metallicity. This revised version was published online in July 2006 with corrections to the Cover Date.     

Collapse and fragmentation of rotating magnetized clouds – II. Binary formation and fragmentation of first cores

Subsequent to Paper I, the evolution and fragmentation of a rotating magnetized cloud are studied with use of three-dimensional magnetohydrodynamic nested grid simulations. After the isothermal runaway collapse, an adiabatic gas forms a protostellar first core at the centre of the cloud. When the isothermal gas is stable for fragmentation in a contracting disc, the adiabatic core often breaks into several fragments. Conditions for fragmentation and binary formation are studied. All the cores which show fragmentation are geometrically thin, as the diameter-to-thickness ratio is larger than 3. Two patterns of fragmentation are found. (1) When a thin disc is supported by centrifugal force, the disc fragments into a ring configuration (ring fragmentation). This is realized in a rapidly rotating adiabatic core as  Ω > 0.2τ⁻¹_ff  , where Ω and  τ_ff  represent the angular rotation speed and the free-fall time of the core, respectively. (2) On the other hand, the disc is deformed to an elongated bar in the isothermal stage for a strongly magnetized or rapidly rotating cloud. The bar breaks into 2–4 fragments (bar fragmentation). Even if a disc is thin, the disc dominated by the magnetic force or thermal pressure is stable and forms a single compact body. In either ring or bar fragmentation mode, the fragments contract and a pair of outflows is ejected from the vicinities of the compact cores. The orbital angular momentum is larger than the spin angular momentum in the ring fragmentation. On the other hand, fragments often quickly merge in the bar fragmentation, since the orbital angular momentum is smaller than the spin angular momentum in this case. Comparison with observations is also shown.     

The stability and evolution of two-component isothermal clusters

The stability of two-component isothermal clusters surrounded by a rigid non-conducting spherical wall is examined by linear normal analyses and nonlinear simulations. The examinations are done for four types of models, classified by the differences concerning gravo-thermal stability and Spitzer's condition. Our results show that perturbations in gravo-thermally stable systems disappears with time and the systems tend to isothermal ones with equipartition, as is expected. On the other hand, in the gravo-thermally unstable systems, the presence of small amount of massive component which has higher central density accelerates the gravo-thermal collapse by heat flowing from the massive component to the less massive component and being transported outward efficiently. This effect of the interaction between two components on gravo-thermal collapse is shown clearly in the forms of the respective eigenfunctions.On leave from Department of Astronomy, Institut Teknologi, Bandung.     

Analysis of Families of Triple Close Approaches Occurring inStellar Systems in Three Dimensions (II)

This is the second paper of a trilogy dealing with the role of triple encounters with low initial velocities and equal masses in the evolution of stellar systems in three dimensional space. It shows how a condition of complete collapse may be perturbed to obtain well-established families of asymmetric triple close approaches with systematic regularity of escape with the formation of a binary. The main result is that when perturbation is introduced two close approaches called the first close approach and the second close approach occur in the same plane but the binary formed and the escaper are not in that plane. Further it is observed that the conjecture of Szebehely (1977) viz. `The measure of escaping orbits is significantly higher than the measure of stable orbits' is likely to be true. The third and last paper offers applications in stellar systems. This revised version was published online in July 2006 with corrections to the Cover Date.     

Neutrinooszillationen in Neutronensternmaterie

According to the suggestion of T. J. Mazurek (1979) neutrino oscillations occuring during the dynamic stellar collapse (M ≥ 10M) could be result in a transfer of leptonic zero-point energy to baryons. Then the adiabatic index increases above γ ≥ 4/3, and such an increase is necessary to reverse the collapse. From the theory of neutrino oscillations of B. Pontekorvo (1967) we derive the oszillation length L of neutrinos in vacuum and the characteristic oscillation lengh L^* of neutrinos taking into consideration the refraction index n_e of neutron star matter. The comparison of both oscillation lenghts shows that for electron densities, characteristically of neutron star matter, the oscillation lenght L is considerable larger than the oscillation lenght L^*. Therefore neutrino oscillations cannot influence the scenario for neutrino emission of the neutron star.     

Grains accretion processes in a proto-planetary nebula

Some mechanisms which are expected to produce the growth of dust grains in the protosolar nebula are studied during the isothermal and the adiabatic phase of the gravitational collapse. Owing to the low sticking efficiency in the grain-grain collisions and also to the impossibility of gas capture by solid particles in the physical environment considered, the main result is the production in about 10⁶ yr of a set of particles similar in mass. The obtained mass limit (10⁻⁸–10⁻⁹ g) depends on the physical properties of the grains, and seems to be independent of the turbulence model used for the gas motion.     

On the Collapse Dynamics of Cold Stars with a Phase Transition in the Interior

Cold stars can become unstable due to a phase transition in the centre. By small perturbations the collapse is triggered toward a new stable configuration. The features of the collapse are studied by means of a parameter-independent model with piece by piece uniform density. Possible consequences of such a collapse for neutron stars are: extreme speed-up, heating and a supernova-like release of binding energy. The dynamical stability properties of the model are considered in the framework of General Relativity. In particular the very beginning of the collapse can be treated by analytical means. In a certain range relativistic effects can prevent a configuration to collapse.     

Numerical study of the adiabatic index effect for the Vishniac instability in supernova remnants

The Vishniac instability is supposed to explain the fragmentation of the thin shell of shocked matter in the radiative phase of supernova remnants. However its implication and its consequence on the morphological evolution of stellar systems is not fully demonstrated. The present paper tackles this subject by numerical simulations and focus on the role of the adiabatic index in the instability growth. The HYDRO-MUSCL 2D hydrodynamics code has been used to simulate the evolution of a supernova remnant thin shell and the triggering of the Vishniac instability in this thin shell. We have studied the temporal behavior of the perturbation. The first result of the numerical study is the existence of the Vishniac instability in the simulations. This result is proved by the overstability process observed in the simulations as predicted by the theoretical analysis. The second important result is the damping of the perturbation at late evolution and for all the set of parameters. Indeed the accretion of matter onto the shock damps the instability when theoretical analysis predicts its occurrence.     

Mass loss and spin

The evolution of spin in a star undergoing adiabatic mass loss is considered. It is established that adiabatic mass loss cannot drive spin. If such a mass loss is due to a binary component overflowing its Roche lobe then this result rules out the possibility of asynchronous rotation in close contact systems. If the mass loss is driven by some other mechanism then a spin up of the star is also not possible.