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After buckling, statically indeterminate trusses, beams, and other strictly symmetric structures may collapse under loads which reach limiting magnitudes. Optimal design is discussed for prescribed values of these collapse loads.
The self-consistent dynamical development of six stellar systems, started from rotating spherical configurations, has been studied by means of a fully three-dimensional n-body integration. The six examples had different initial angular velocities and velocity dispersions. All settled down into prolate bars rotating about a short axis within two initial rotation periods. The bars are long-lived, robust, and stable. Bars are the natural form toward which rapidly rotating stellar dynamical systems develop, instead of the flattened axisymmetric disks that had been expected. The early stages of each collapse are reasonably well described by a theoretical model according to which a collapse passes through a sequence of rigidly rotating, uniform-density spheroids. The first significant departures from spheroidal form were axisymmetric in all cases. Rings formed in some examples, sheets in others, with transition cases between these extremes. Nonaxisymmetric forms developed from these intermediate stages.
Although 1-D (spherically symmetric) experiments of protostar collapse are highly idealized, they are the only ones which have been carried to a stage where a 'stellar' object is formed. Experiments have shown that the parameters (e.g., radius and luminosity) of the visible stellar core are sensitive to the assumed initial conditions, particularly the initial density. One of the major findings of 2-D numerical experiments is the formation of rings. Three-dimensional hydrodynamical calculations indicate that a collapsing cloud will break up into two or more orbiting subcondensations with the possible subsequent development of a stellar multiple system.
The paper presents the results of a numerical calculation of the collapse of an idealized protostellar gas cloud including the effects of a 'frozen-in' magnetic field. The 'traditional' picture of magnetic effects on gas clouds and recent observational and theoretical work on the subject are summarized. Attention is given to the method of calculation and the results are interpreted. It is found that the central magnetic field in the collapsing cloud model follows a rho to the 1/2 power relation, and the discussion implies that this is a general result which should hold true for some range of initial conditions around those chosen. In addition, it is found that the outer envelope of the cloud will be held up by tension in the field lines.
Results of the first self-consistent numerical calculations of the dynamic collapse of a magnetized protostellar gas cloud are presented. Symmetry about an axis parallel to the initial magnetic field direction has been assumed, so that the calculations could be performed on a two-dimensional grid. Also, the cloud was taken to be nonrotating and isothermal, and the magnetic field was assumed to remain frozen in to the gas. As starting models for the calculations, gas spheres with uniform density and magnetic field were used. The time evolution of the clouds has been calculated for roughly two initial free-fall times, at which point the central density has increased by a factor of approximately 10,000 to 1,000,000. Several such calculations have been performed for different values of the cloud's initial thermal, magnetic, and gravitational energies. In virtually all cases it is found that, once a flattened core forms in the cloud, the central magnetic field strength, B, varies with gas density, rho, according to (d log B/d log rho) = 1/2. This behavior is independent of the initial energy ratios mentioned above. It is also found that the magnetic field is able to prevent completely the collapse of part of the outer envelope of the cloud.
Observations are presented of the parametric decay and spatial collapse of Langmuir waves driven by an electron beam streaming into the solar wind from the Jovian bow shock. Long wavelength Langmuir waves upstream of the bow shock are effectively converted into short wavelength waves no longer in resonance with the beam. The conversion is shown to be the result of a nonlinear interaction involving the beam-driven pump, a sideband emission, and a low level of ion-acoustic turbulence. The beam-driven Langmuir wave emission breaks up into a complex sideband structure with both positive and negative Doppler shifts. In some cases, the sideband emission consists of isolated wave packets with very short duration bursts, which are very intense and are thought to consist of envelope solitons which have collapsed to spatial scales of only a few Debye lengths.
Levine's (1974) concept of a collapsing magnetic neutral sheet which can accelerate ambient protons to several times the mean thermal speed, provided that the collapse time scale is shorter than the proton Coulomb loss time, is applied to heavier elements in an investigation of the composition of the particles which can be accelerated by such a sheet. Tables of ionization equilibrium are combined with the thermal structure of a constant pressure loop in order to calculate phi sub x (the fraction of an element x which is accelerated) of 18 heavy elements, from carbon to nickel which satisfy a maximum-ionization criterion. Normalizing phi sub x to oxygen, it is found that relative to the composition of the ambient materials, C and N can be depleted by factors of up to 2-10, while other heavy elements, along with hydrogen, are enhanced. The numerical results obtained are qualitatively similar to anomalous abundances reported among solar flare particles.
In flares occurring in magnetic neutral sheets, the collapsing of the magnetic fields toward the neutral sheet results in their acting as mirrors in a first-order Fermi acceleration process and constitutes an intrinsic source of particle pre-acceleration. Although modest, the acceleration injects particles into the main flare acceleration with a composition which is different from that of the ambient corona. Upon calculating the fractions of 18 elements which are accelerated in the collapsing mirrors, covering the series from C to N, and then normalizing to O, peak abundances are found at Si and Fe, with local minima at N and in the S-Ca region. There is a qualitative similarity between these relative abundances and the relative enhancement factors of the elements among solar flare particles.
Wave-wave interactions are examined, particularly for significance in relation to Langmuir turbulence associated with type III solar radio emissions. The Zakharov equations are considered as two coupled nonlinear partial differential equations for the low-frequency density and the Langmuir envelope, and the electron and ion temperatures are regarded as equal. The Zhakharov equations are solved in one dimension, and display no collapse of a driven broadband pump, indicating that scattering instabilities are important in the saturation of a beam-plasma instability. A two-dimensional numerical simulation is developed to interpret the role of parametric instabilities, and it is concluded that the distortion of a wave packet is due to scattering instability, and not a change in the physics of direct collapse due to an artificial numerical damping. Further discussions are presented for comparisons of models of Langmuir turbulence in type III bursts.
The evolutions of the envelopes of collapsing, accreting, isothermal clouds have been numerically calculated for both spherically symmetric and rotating (axisymmetric) clouds. The results provide a cohesive picture of isothermal collapse, and their relationship to previous numerical calculations and similarity solutions is discussed. Even with a large initial rotation rate, the majority of the cloud envelope is accreted, in one case leaving behind a large-scale circulation current. The calculations are performed for both initially uniform density and centrally condensed clouds. Density and velocity profiles for a wide variety of observed systems are compared with those obtained in this study, providing a preliminary assessment of the stage of evolution and initial structure for the observed systems.
Solid rocket booster cavity collapse flight measurements included external pressures on the motor case and aft skirt, internal motor case pressures, accelerometers located in the forward skirt, mid-body area, and aft skirt, as well as strain gages located on the skin of the motor case. This flight data yielded applied pressure longitudinal and circumferential distributions which compare well with model test predictions. The internal motor case ullage pressure, which is below atmospheric due to the rapid cooling of the hot internal gas, was more severe (lower) than anticipated due to the ullage gas being hotter than predicted. The structural dynamic response characteristics were as expected. Structural ring and wall damage are detailed and are considered to be attributable to the direct application of cavity collapse pressure combined with the structurally destabilizing, low internal motor case pressure.
Radiative transfer in the Eddington approximation is included in a multidimensional, self-gravitational, hydrodynamical computer code. Details of the numerical solution and thermodynamic relations are given. Comparison calculations with previous spherically symmetrical models of protostellar collapse are used to validate the basic approach and the artifices which allow the explicit hydrodynamics code to follow the accretion of gas onto a quasi-equilibrium core. A series of axisymmetric models is used to investigate the importance of rotation in collapsing clouds, as the initial amount of angular momentum is lowered, with an emphasis on the possible formation of rings. Rings readily form even in the nonisothermal regime except for very low initial angular momenta; even these clouds may experience ring formation prior to reaching stellar densities. The models imply that other effects (such as gravitational torques or turbulent viscosity) may be necesary to avoid binary formation and thus result in a presolar nebula consistent with the assumptions of either Safronov or Cameron.
A generalized model which accounts for the effects of initially uniform and slow rotation is defined for the spherical collapse of a singular isothermal sphere such as protosolar and binary nebulae. An initial unstable equilibrium state is described for a sound speed of 0.35 km/sec and a rotation rate of 10 to the -14th/sec for the molecular cloud surrounding the accreting core. The total angular momentum and mass of the inner cloud is set equal to solar system values. The evolution of the collapse is traced by applying a perturbation analysis to the similarity solution for a nonrotating condition, and matched asymptotic expansions solve the hydrodynamic equations. The model is concluded a valid tool for studying star and nebular disk formation.
The astrophysical site of r-process nucleosynthesis is investigated theoretically using two-dimensional expanding-vortex stellar-collapse simulations based on the Eulerian adaptive-mesh MHD code of Symbalisty (1984). The results are presented graphically, and it is found that the classical r-process can be explained as the result of the collapse of the highly rotating iron core of a 15-solar-mass star, in which the angular velocity along the rotation axis reaches a maximum of 1000 rad/s, corresponding to an angular momentum of 3.4 x 10 to the 48th erg/s for the inner 1.5 solar mass. The ejected jets are shown to yield about 0.0004 solar mass per supernova, sufficient to explain the observed abundances of r-process products.
The problem of reflection of magnetohydrodynamic (MHD) waves at the boundary of a numerical grid has to be resolved in order to obtain reliable results for the end state of the (isothermal) collapse of a rotating, magnetic protostellar cloud. Since the goal of investigating magnetic braking in collapse simulations is to see if the transport of angular momentum via alfven waves is large enough to solve the angular momentum problem an approximation that artificially suppresses large amplitudes in the MHD waves can be self-defeating. For this reason, four alternate methods of handling reflected waves where no assumptions are made regarding the amplitudes of the waves were investigated. In order to study this problem (of reflection) without interference from other effects these methods were tried on two simpler cases. The four methods are discussed.
Numerical calculations are presented for rigorous models spanning a four-dimensional parameter space of initial conditions of the three-dimensional collapse of rotating protostellar clouds, encompassing radiative transfer in the Eddington approximation and detailed thermodynamical relations. It is found that protostellar formation may involve a few stages of hierarchical fragmentation terminated by increased thermal pressure in the nonisothermal regime, that high thermal energy clouds remain nearly axisymmetric during the first dynamic collapse phase, and that very slowly rotating clouds can fragment. The presolar nebula was probably formed from a cloud with very little initial rotation.
Rigorous numerical models of the collapse and accretion of rotating, axisymmetric, isothermal interstellar clouds are studied. The results show that molecular gas disks and evacuated bipolar cavities both appear to be natural consequences of the collapse of rotating interstellar clouds. Dynamically significant magnetic fields may not be necessary for explaining either phenomenon. The models strongly support theoretical models of the type where an isotropic wind from a pre-main sequence star is extrinsically collimated by a rotationally derived molecular gas cloud. The models imply that collimation should be strongest on small scales where rotational effects are most important, i.e., in the dense region of the molecular gas disk.
The density at which electron capture can ignite oxygen burning in 8-10 solar mass stars is investigated within the uncertainties of the semiconvection theory. The Ledoux criterion for convection is adopted in order to take into account the maximum effect of heating due to electron capture. It is found that oxygen burning is not ignited by electron captures on Mg-24 and Na-24 at the central density of about 4 x 10 to the 9th g/cu cm, but by capture on Ne-20 at about 1 x 10 to the 10th g/cu cm. This ignition density is 2.5 times lower than in the case with the Schwarzschild criterion, but is still high enough to induce a collapse by electron capture rather than a total explosion of the core. When the oxygen-burning front arrives at M(r) of roughly 0.3 solar mass, the contraction reaches a quasi-dynamic collapse.