Jump relations across a shock in non-ideal gas flow

Generalized forms of jump relations are obtained for one dimensional shock waves propagating in a non-ideal gas which reduce to Rankine-Hugoniot conditions for shocks in idea gas when non-idealness parameter becomes zero. The equation of state for non-ideal gas is considered as given by Landau and Lifshitz. The jump relations for pressure, density, temperature, particle velocity, and change in entropy across the shock are derived in terms of upstream Mach number. Finally, the useful forms of the shock jump relations for weak and strong shocks, respectively, are obtained in terms of the non-idealness parameter. It is observed that the shock waves may arise in flow of real fluids where upstream Mach number is less than unity.     

Shock jump relations for a dusty gas atmosphere

This paper presents simplified forms of jump relations for one dimensional shock waves propagating in a dusty gas. The dusty gas is assumed to be a mixture of a perfect gas and spherically small solid particles, in which solid particles are continuously distributed. The simplified jump relations for the pressure, the temperature, the density, the velocity of the mixture and the speed of sound have been derived in terms of the upstream Mach number. The expressions for the adiabatic compressibility of the mixture and the change-in-entropy across the shock front have also been derived in terms of the upstream Mach number. Further, the handy forms of shock jump relations have been obtained in terms of the initial volume fraction of small solid particles and the ratio of specific heats of the mixture, simultaneously for the two cases viz., (i) when the shock is weak and, (ii) when it is strong. The simplified shock jump relations reduce to the Rankine-Hugoniot conditions for shock waves in an ideal gas when the mass fraction (concentration) of solid particles in the mixture becomes zero. Finally, the effects due to the mass fraction of solid particles in the mixture, and the ratio of the density of solid particles to the initial density of the gas are studied on the pressure, the temperature, the density, the velocity of the mixture, the speed of sound, the adiabatic compressibility of the mixture and the change-in-entropy across the shock front. The results provided a clear picture of whether and how the presence of dust particles affects the flow field behind the shock front. The aim of this paper is to contribute to the understanding of how the shock waves behave in the gas-solid particle two-phase flows.     

A model of the thermal processing of particles in solar nebula shocks: Application to the cooling rates of chondrules

Abstract— We present a model for the thermal processing of particles in shock waves typical of the solar nebula. This shock model improves on existing models in that the dissociation and recombination of H₂ and the evaporation of particles are accounted for in their effects on the mass, momentum and energy fluxes. Also, besides thermal exchange with the gas and gas‐drag heating, particles can be heated by absorbing the thermal radiation emitted by other particles. The flow of radiation is calculated using the equations of radiative transfer in a slab geometry. We compute the thermal histories of particles as they encounter and pass through the shock. We apply this shock model to the melting and cooling of chondrules in the solar nebula. We constrain the combinations of shock speed and gas density needed for chondrules to reach melting temperatures, and show that these are consistent with shock waves generated by gravitational instabilities in the protoplanetary disk. After their melting, cooling rates of chondrules in the range 10–1000 K h^?1 are naturally reproduced by the shock model. Chondrules are kept warm by the reservoir of hot shocked gas, which cools only as fast as the dust grains and chondrules themselves can radiate away the gas's energy. We predict a positive correlation between the concentration of chondrules in a region and the cooling rates of chondrules in that region. This correlation is supported by the unusually high frequency of (rapidly cooled) barred chondrules among compound chondrules, which must have collided preferentially in regions of high chondrule density. We discuss these and other compelling consistencies between the meteoritic record and the shock wave model of chondrule formation.     

The post-explosion shock propagation in the central region of our Galaxy

Immediate consequences of nuclear explosions on the structure and physical state of a galactic disk are considered in this paper. Explosions in the nucleus of a Galaxy generate strong shock waves which, when propagating onward heat and condensing the gas, form thin dense ring-like gaseous features behind it. Such rings and dense gaseous complexes have been observed in the central region of the Galaxy. These features have been treated here as the remnants of galactic shocks generated by nuclear explosions. We have estimated the time elapsed since the corresponding explosion, the energy released by explosion and the initial temperature and the velocity of the shock wave thus generated. The cooling of the gas heated by strong shocks has also been considered. The time taken by shock-heated gas to cool to its original temperature has been estimated to be of the order of 10⁵ to 10⁶ yr, according to the initial shock temperature which is about 9×10⁶ K or 6.4×10⁷ K. The rate of emission of energy and the total amount of energy dissipated away in the form of radiation in the cooling process, have been calculated for different values of initial shocktemperatures and also for different field intensities. The high-energy radiation emitted in the cooling process is suggested here as a source for the heating of dust grains, which ultimately are radiated in the infrared spectrum. Thus, a part of the infrared radiation, as measured by many authors, in the central region of the Galaxy, may originate ultimately from the cooling of the shock-heated gas there.     

The weak shock in the core of the Perseus cluster

The dissipation of energy from sound waves and weak shocks is one of the most promising mechanisms for coupling active galactic nucleus (AGN) activity to the surrounding intracluster medium, and so offsetting cooling in cluster cores. We present a detailed analysis of the weak shock found in deep Chandra observations of the Perseus cluster core. A comparison of the spectra either side of the shock front shows that they are very similar. By performing a deprojection analysis of a sector containing the shock, we produce temperature and density profiles across the shock front. These show no evidence for a temperature jump coincident with the density jump. To understand this result, we model the shock formation using 1D hydrodynamic simulations including models with thermal conduction and  γ < 5/3  gas. These models do not agree well with the data, suggesting that further physics is needed to explain the shock structure. We suggest that an interaction between the shock and the Hα filaments could have a significant effect on cooling the post-shock gas.
We also calculate the thermal energy liberated by the weak shock. The total energy in the shocked region is about 3.5 times the work needed to inflate the bubbles adiabatically, and the power of the shock is around  6 × 10⁴⁴ erg s⁻¹  per bubble, just over  10⁴⁵ erg s⁻¹  in total.     

Formation of planetary systems by successive contractions of the solar nebula

The theory discussed in the present paper is a solar nebula-type theory which assumes the initial existence of a big disk-shaped gas cloud in rotational motion around the Sun. At the outer edge of the gas cloud there is a steady loss of angular momentum, which is mainly caused by the diffusion induced by turbulence and shock waves. This leads to the formation of a doughnutshaped gas ring at the edge of the cloud, outside of which there is plasma in a state of partial corotation. The gas ring is then slowly shifted towards the Sun, whereby the grains of solid matter within the gas cloud are also transported and collected within the gas torus. During the contraction process the following two situations arise: First, due to the small amount of friction, the angular momentum of the inner part of the ring rapidly exceeds that of the outer part. Second, the angle between the orbits of the inner and outer part of the gas ring increases gradually. When, during contraction, a certain distance is covered, the gas ring turns over, i.e. there is a sudden interchange of the inner and outer parts of the gas ring, where two adjacent rings of solid matter (jet streams) are formed. Immediately after the turn-over process the speed of contraction is at first drastically reduced, but then the gas ring is shifted once more towards the Sun. This process is then repeated periodically. The planets originate from the outer rings of solid matter, which contain much more matter than their adjacent inner rings. The inclination between the inner and outer rings is roughly 5°. In particular, Mercury, the Moon, Titan as well as Triton result from the innermost rings of matter. Having gone through the formation process, most of the planets acquire a rotating gas disk out of which the regular satellites are also created by the same periodic contraction process (hetegonic principle). This theory is the first that can explain all noteworthy facts about our planetary system and the satellite systems in a qualitative yet conclusive way.     

Atmospheric shock waves after impacts of cosmic bodies up to 1000 m in diameter

The results of numerical modeling of meteoroids' interaction with Earth's atmosphere are presented. We model the entry in two dimensions and then interpolate the results into a 3‐D model to calculate interaction of shock waves with the surface. Maximum shock pressures, wind speeds, and areas subjected to substantial overpressure are calculated for oblique impacts of asteroids and comets. We show that vertical impacts produce a smaller damage zone on the surface than oblique ones. Damage caused by shock waves covers an order of magnitude larger area than any other hazardous effects. The function of energy release in the atmosphere, which is traditionally used in meteoritics, has a limited application if cosmic bodies are larger than tens of meters in diameter: at each time moment energy is smoothed along a substantial length of the trajectory; both emitted radiation (routinely used for calibration of semi‐analytical models) and shock wave amplitude are complex functions of temperature–density distributions in atmosphere.     

Nonisothermal Wave Motions in the HII Regions

The propagation of the linear and nonlinear waves in the HII regions is considered. The relaxation processes to the temperature and ionization fraction are taken into account. It is shown, that the essential damping of the waves takes place. The motion at long wavelengths is not isothermal and the velocity perturbations should be accompanied with the noticeable temperature variations. Instability of the traveling waves in the partially ionized hydrogen is possible. This effect may be important in the gas layer between ionization and shock fronts. It is found out, that the nonlinear waves can be generated by the ionization-shock front. This revised version was published online in July 2006 with corrections to the Cover Date.     

In-shock cooling in numerical simulations

We model a one-dimensional shock-tube using smoothed particle hydrodynamics and investigate the consequences of having finite shock-width in numerical simulations caused by finite resolution of the codes. We investigate the cooling of gas during passage through the shock for three different cooling regimes.
For a theoretical shock temperature of 10⁵ K, the maximum temperature of the gas is much reduced. When the ratio of the cooling time to shock-crossing time was 8, we found a reduction of 25 per cent in the maximum temperature reached by the gas. When the ratio was reduced to 1.2, the maximum temperature reached dropped to 50 per cent of the theoretical value. In both cases the cooling time was reduced by a factor of 2.
At lower temperatures, we are especially interested in the production of molecular hydrogen, and so we follow the ionization level and H₂ abundance across the shock. The effect of in-shock cooling is substantial: the maximum temperature the gas reaches compared with the theoretical temperature is found to vary between 0.15 and 0.81, depending upon the shock strength and mass resolution. The downstream ionization level is reduced from the theoretical level by a factor of between 2.4 and 12.5, and the resulting H₂ abundance by a factor of 1.35 to 2.22.
At temperatures above 10⁵ K, radiative shocks are unstable and will oscillate. We find that the shock jump temperature varies by a factor of 20 because of these oscillations.
We conclude that extreme caution must be exercised when interpreting the results of simulations of galaxy formation.     

Diagnostics of supernova remnant shock waves

Fairly simple models can explain the emission from non-radiative shock waves in supernova remnants. This talk reviews some of the more robust diagnostics of shock parameters and some of the implications for the physics of collisionless shock waves in interstellar gas. The Hopkins Ultraviolet Telescope observed several non-radiative shocks during the ASTRO-2 mission, and the spectra have interesting implications for the physics of collisionless shocks.     

Star formation induced by supersonic turbulence in interstellar molecular cloud

We studied fragmentation process of the interstellar molecular cloud which is predominated by supersonic turbulence with special regard to collisions of turbulent gas elements and formation of a shock-compressed layer by receding shock waves. The propagation of the shock waves and the evolution of the compressed layer are followed by one-dimensional gas dynamical simulation until self-gravity becomes significant, taking account of the effects of thermal properties of the molecular gas and magnetic fields. It is shown that the efficient cooling by CO molecules and its sensitive dependence on gas density make the shock-compressed layer so cold and dense that the layer becomes gravitationally unstable and breaks into fragments even if the gas elements are gravitationally stable prior to the collision. The mass of the unstable fragment is estimated to be about two solar masses or less, irrespective of the presence of the magnetic field. The stars formed by collisions of supersonic turbulent gas elements accelerate the surrounding gas in T Tauri stage and replenish the turbulent energy to maintain the mechanical equilibrium of the molecular cloud.     

The formation of shock waves in the recombination era of the universe

The formation and the evolution of shock waves in a two-component fluid during the recombination era of the Universe is studied. The fluid consists of photons and ionized hydrogen gas with interaction due to Thomson scattering. The analysis uses the expansion procedure in powers of the coordinates near the travelling wave front. Incorporating discontinuities in the first derivatives of the physical quantities, we are able to give the analytical solution which answers the questions as to when the discontinuities appear in the physical quantities themselves. Of the two possible waves (a radiation- and a matter-dominated wave) only the matter-dominated wave shows the shock phenomena. However, the shock is induced by the radiation due to Thomson scattering. Therefore, the time for the shock formation depends on the initial amplitude of the wave (as in usual shock phenomena) and on the collision frequency of the photons.     

Jump relations for magnetohydrodynamic shock waves in non-ideal gas flow

The generalized jump relations across the magnetohydrodynamic (MHD) shock front in non-ideal gas are derived considering the equation of state for non-ideal gas as given by Landau and Lifshitz. The jump relations for pressure, density, and particle velocity have been derived, respectively in terms of a compression ratio. Further, the simplified forms of the MHD shock jump relations have been obtained in terms of non-idealness parameter, simultaneously for the two cases viz., (i) when the shock is weak and, (ii) when it is strong. Finally, the cases of strong and weak shocks are explored under two distinct conditions viz., (i) when the applied magnetic field is strong and, (ii) when the field is weak. The aim of this paper is to contribute to the understanding of how shock waves behave in magnetized environment of non-ideal gases.     

Shock structures in dusty plasma in the presence of strong electrostatic interaction

The fluid approach is employed to investigate theoretically the effect of strong electrostatic interaction on the dust acoustic (DA) shock waves near to the liquid-crystal phase transition in strongly coupled dusty plasma. The strong electrostatic interaction is modeled by effective electrostatic temperature which is considered as a dynamical variable. It is shown that the nonlinear evolution of dust acoustic shock waves in the present model is governed by a Burger equation, the coefficients in which are modified by strong coupling effect. Then, it is shown that how the perturbation of the effective electrostatic temperature modifies the basic properties of the DA shock waves.     

Emission stratification in two-component HII regions with stellar wind

To explain the variety of observed optical emission stratification in the shells around Wolf-Rayet stars, we have calculated the nonstationary cooling of a homogeneous gas layer heated to a temperature (0.4–2) × 10⁵ K. We have assumed that the nebula is ionized by its central star and consists of a rarefied gas and a set of clouds with different densities through which adiabatic shock waves produced by the stellar wind propagate. Based on this model, we have determined the sequence in which the emission in Hα and in nebular oxygen lines appears. The Hα emission attributable to the electron-collision excitation of hydrogen atoms is produced earliest on the periphery of nebulae, the [O III] line emission follows next, and, finally, the Hα recombination emission is produced. The results obtained are in good agreement with the observational data.     

Shock waves with large energy losses by direct radiation from the front

Highly nonadiabatic shock waves are formed at an early stage of a supernova explosion inside a stellar wind because of the large energy losses by direct radiation from the front. The properties of such waves are considered for velocities of (5?25)×10³km s^?1 and gas densities of 10^?17?10^?10 g cm^?3. A critical energy flux going to “infinity” that separates two modes is shown to exist. If the flux is lower than the critical one, then energy losses cause even an increase in the post-shock temperature. An excess of the flux over its critical value results in an abrupt cooling and in a strong compression of the gas. For the flux equal to the critical one, the post-shock gas velocity matches the isothermal speed of sound. Approximate formulas are given for estimating the degree of gas compression and the post-shock radiation-to-gas pressure ratio at energy losses equal to the critical ones and for the limiting compression.     

Cross-scale coupling at a perpendicular collisionless shock

A full particle simulation study is carried out on a perpendicular collisionless shock with a relatively low Alfven Mach number (M_A = 5). Recent self-consistent hybrid and full particle simulations have demonstrated ion kinetics are essential for the non-stationarity of perpendicular collisionless shocks, which means that physical processes due to ion kinetics modify the shock jump condition for fluid plasmas. This is a cross-scale coupling between fluid dynamics and ion kinetics. On the other hand, it is not easy to study cross-scale coupling of electron kinetics with ion kinetics or fluid dynamics, because it is a heavy task to conduct large-scale full particle simulations of collisionless shocks. In the present study, we have performed a two-dimensional (2D) electromagnetic full particle simulation with a “shock-rest-frame model”. The simulation domain is taken to be larger than the ion inertial length in order to include full kinetics of both electrons and ions. The present simulation result has confirmed the transition of shock structures from the cyclic self-reformation to the quasi-stationary shock front. During the transition, electrons and ions are thermalized in the direction parallel to the shock magnetic field. Ions are thermalized by low-frequency electromagnetic waves (or rippled structures) excited by strong ion temperature anisotropy at the shock foot, while electrons are thermalized by high-frequency electromagnetic waves (or whistler mode waves) excited by electron temperature anisotropy at the shock overshoot. Ion acoustic waves are also excited at the shock overshoot where the electron parallel temperature becomes higher than the ion parallel temperature. We expect that ion acoustic waves are responsible for parallel diffusion of both electrons and ions, and that a cross-scale coupling between an ion-scale mesoscopic instability and an electron-scale microscopic instability is important for structures and dynamics of a collisionless perpendicular shock.     

Explosion-triggered star formation in the central region of the Galaxy

The role of shock waves generated by explosions in the nucleus of a galaxy and propagating through the surrounding gas, on the efficiency of star formation in this region, is investigated. It is found that if the gas in the shock-compressed layer follows polytropic changes in the density and temperature, a very wide range of masses of the fragments can be formed. Calculation shows that under suitable physical conditions a burst of star formation is actually realized.     

Grain destruction in interstellar shock waves

Interstellar shock waves can erode and destroy grains present in the shocked gas, primarily as the result of sputtering and grain-grain collisions. Uncertainties in current estimates of sputtering yields are reviewed. Results are presented for the simple case of sputtering of fast grains being stopped in cold gas. An upper limit is derived for sputtering of refractory grains in C-type MHD shocks: shock speedsv _s 50 km s^–1 are required for return of more than 30% of the silicate to the gas phase. Sputtering can also be important for removing molecular ice mantles from grains in two-fluid MHD shock waves in molecular gas. Recent estimates of refractory grain lifetimes against destruction in shock waves are summarized, and the implications of these short lifetimes are discussed.     

Propagation of plane relativistic shock waves

Similarity solutions for propagation of plane shock waves in a relativistic gas, where the nucleon number density varies linearly as distance from its edge at near vacuum is obtained. The shock front moves with constant speed and the solutions are applicable only to an isothermal medium or cold gas.