Analytical structure of steady radiative shocks in magnetic cataclysmic variables

In this paper we study specific classes of radiating shocks which are widely spread in astrophysical environments. We present new analytical solutions available for any exponents of typical power-law cooling function which generalize the five special cases (corresponding to specific values of these exponents) of radiating shocks structure and proceed to the analytical determination of physical quantities. Then an application of the results for the accretion shock in polar is realized. A discussion of the reproducibility of cooling layer in laboratory is proposed using scaling laws.     

On the linear analysis of unstable radiative shocks

We study the stability properties of strong hydrodynamic shocks and their associated radiative cooling layers. We explore a range of conditions which covers both molecular and atomic gas impacting against a rigid wall. Through a linear analysis employing a cooling function of the form  Λ∝ρ^β T ^α  and a specific heat ratio of γ, we determine the overstability regime in the parameter space consisting of  α, β  and γ. In general, if α is sufficiently low, the fundamental mode leads to long-wavelength growing oscillations. For the fundamental mode, we find that values of γ corresponding to molecular hydrodynamics lead to a significantly restricted instability range for α in comparison with the shocks in a monatomic medium. The conditions for the growth of higher-order modes, however, are relatively unchanged. This predicts that certain molecular shocks are prone to displaying signatures of small-scale rapid variability. Dissociative shocks, however, can be subject to a large-scale overstability if subsequent molecule formation in the cooling layer abruptly increases the cooling rate. In contrast to the dynamical rippling overstability, the cooling overstability is suppressed for a sufficiently low specific heat ratio.     

Classification of and recent research involving radiative shocks

Radiative shocks (RS) occur in astrophysical systems and in high-energy density laboratory experiments. Aided by three dimensionless parameters, we propose a classification of RS into four types, integrating previous work that has focused independently on optical depth and on Mach number. Specific terms, such as a cooling function, a radiative flux, or radiative energy and pressure must be added to the Euler equations in order to model these various kinds of shocks. We examine how these terms correspond to the radiative classification regimes. In astrophysics, observed RS arise generally in optically thin material. Thus, radiation escapes without interaction with the surrounding gas, except perhaps to ionize it, and the energy loss in such shocks can be modeled by a cooling function Λ. In this case only the post-shock region is structured by the radiation cooling. We found the analytical solution for hydrodynamic equations including Λ∝ ρ ^ε P ^ζ x ^θ for arbitrary values of ε, ζ, θ. This is a completely new result. An application of this calculation for the accretion shock in cataclysmic variables of polar type is given in astrophysical terms. We also draw a parallel between RS experiments performed using the LULI2000 laser facility, in France and the Omega laser Facility, in USA. RS developed in these laboratories are more or less optically thick. These high-Mach number RS present a radiative precursor.     

The influence of the Mach number on the stability of radiative shocks

We study the stability properties of hydrodynamic shocks with finite Mach numbers. The linear analysis supplements previous analyses which took the strong shock limit. We derive the linearized equations for a general specific heat ratio as well as temperature and density power-law cooling functions, corresponding to a range of conditions relevant to interstellar atomic and molecular cooling processes. Boundary conditions corresponding to a return to the upstream temperature  ( R = 1)  and to a cold wall  ( R = 0)  are investigated. We find that for Mach number   M > 5  , the strong shock overstability limits are not significantly modified. For   M < 3  , however, shocks are considerably more stable for most cases. In general, as the shock weakens, the critical values of the temperature power-law index (below which shocks are overstable) are reduced for the overtones more than for the fundamental, which signifies a change in basic behaviour. In the   R = 0  scenario, however, we find that the overstability regime and growth rate of the fundamental mode are increased when cooling is under local thermodynamic equilibrium. We provide a possible explanation for the results in terms of a stabilizing influence provided downstream but a destabilizing effect associated with the shock front. We conclude that the regime of overstability for interstellar atomic shocks is well represented by the strong shock limit unless the upstream gas is hot. Although molecular shocks can be overstable to overtones, the magnetic field provides a significant stabilizing influence.     

Oscillatory instability of radiative shocks with multiple cooling processes

The stand-off shock formed in the accretion flow on to a stationary wall, such as the surface of a white dwarf, may be thermally unstable, depending on the cooling processes which dominate the post-shock flow. Some processes lead to instability, while others tend to stabilize the shock. We consider competition between the destabilizing influence of thermal bremsstrahlung cooling, and a stabilizing process which is a power law in density and temperature. Cyclotron cooling and processes which are of order 1, 3/2 and 2 in density are considered. The relative efficiency and power-law indices of the second mechanism are varied, and particular effects on the stability properties and frequencies of oscillation modes are examined.     

Perturbative analysis of two-temperature radiative shocks with multiple cooling processes

The structure of the hot downstream region below a radiative accretion shock, such as that of an accreting compact object, may oscillate because of a global thermal instability. The oscillatory behaviour depends on the functional forms of the cooling processes, the energy exchanges of electrons and ions in the shock-heated matter, and the boundary conditions. We analyse the stability of a shock with unequal electron and ion temperatures, where the cooling consists of thermal bremsstrahlung radiation which promotes instability, plus a competing process which tends to stabilize the shock. The effect of transverse perturbations is considered also. As an illustration, we study the special case in which the stabilizing cooling process is of order 3/20 in density and 5/2 in temperature, which is an approximation for the effects of cyclotron cooling in magnetic cataclysmic variables. We vary the efficiency of the second cooling process, the strength of the electron–ion exchange and the ratio of electron and ion pressures at the shock, to examine particular effects on the stability properties and frequencies of oscillation modes.     

X-rays from massive OB stars: thermal emission from radiative shocks

Chandra grating spectra of a sample of 15 massive OB stars were analysed under the basic assumption that the X-ray emission is produced in an ensemble of shocks formed in the winds driven by these objects. Shocks develop either as a result of radiation-driven instabilities or due to confinement of the wind by a relatively strong magnetic field, and since they are radiative, a simple model of their X-ray emission was developed that allows a direct comparison with observations. According to our model, the shock structures (clumps, complete or fractional shells) eventually become 'cold' clouds in the X-ray sky of the star. As a result, it is expected that for large covering factors of the hot clumps, there is a high probability for X-ray absorption by the 'cold' clouds, resulting in blueshifted spectral lines. Our analysis has revealed that such a correlation indeed exists for the considered sample of OB stars. As to the temperature characteristics of the X-ray emission plasma, the studied OB stars fall in two groups: (i) one with plasma temperature limited to ∼0.1–0.4 keV and (ii) the other with X-rays produced in plasmas at considerably higher temperatures. We argue that the two groups correspond to different mechanisms for the origin of X-rays: in radiation-driven instability shocks and in magnetically confined wind shocks, respectively.     

Self-similar propagation of axisymmetric radiative magnetogasdynamic shocks with increasing energy,II

In this paper self-similar solutions have been investigated for the propagation of axisymmetric radiative gasdynamic shocks caused by an explosion into an inhomogeneous ideal gas permeated by a current free azimuthal magnetic field. The effects of radiation flux and magnetic field together have been seen in the region of interest on the other flow variables. The total energy of the flow between the inner expanding surface and the shock is taken to be dependent on shock radius obeying a power law. The radiative pressure and energy have been neglected.     

Radio signature of cosmological structure formation shocks

In the course of the formation of cosmological structures, large shock waves are generated in the intracluster medium (ICM). In analogy to processes in supernova remnants, these shock waves may generate a significant population of relativistic electrons which, in turn, produce observable synchrotron emission. The extended radio relics found at the periphery of several clusters and possibly also a fraction of radio halo emission may have this origin. Here, we derive an analytic expression for (i) the total radio power in the downstream region of a cosmological shock wave, and (ii) the width of the radio-emitting region. These expressions predict a spectral slope close to −1 for strong shocks. Moderate shocks, such as those produced in mergers between clusters of galaxies, lead to a somewhat steeper spectrum. Moreover, we predict an upper limit for the radio power of cosmological shocks. Comparing our results to the radio relics in Abell 115, 2256 and 3667, we conclude that the magnetic field in these relics is typically at a level of 0.1 μG. Magnetic fields in the ICM are presumably generated by the shocks themselves; this allows us to calculate the radio emission as a function of the cluster temperature. The resulting emissions agree very well with the radio power–temperature relation found for cluster haloes. Finally, we show that cosmic accretion shocks generate less radio emission than merger shock waves. The latter may, however, be detected with upcoming radio telescopes.     

Stability analyses of two-temperature radiative shocks: formulation, eigenfunctions, luminosity response and boundary conditions

We present a general formulation for stability analyses of radiative shocks with multiple cooling processes, longitudinal and transverse perturbations, and unequal electron and ion temperatures. Using the accretion shocks of magnetic cataclysmic variables as an illustrative application, we investigate the shock instabilities by examining the eigenfunctions of the perturbed hydrodynamic variables. We also investigate the effects of varying the condition at the lower boundary of the post-shock flow from a zero-velocity fixed wall to several alternative types of boundaries involving the perturbed hydrodynamic variables, and the variations of the emission from the post-shock flow under different modes of oscillations. We found that the stability properties for flow with a stationary-wall lower boundary are not significantly affected by perturbing the lower boundary condition, and they are determined mainly by the energy-transport processes. Moreover, there is no obvious correlation between the amplitude or phase of the luminosity response and the stability properties of the system. Stability of the system can, however, be modified in the presence of transverse perturbation. The luminosity responses are also altered by transverse perturbation.     

The effect of heavy ions on the formation and structure of cometary bow shocks

We investigate the interaction of heavy cometary ions with the solar wind and the formation of a bow shock in front of a comet by means of a hybrid (particle ion, fluid electron) simulation code that solves self-consistently for the electromagnetic fields and the motion of the charged particles. This kinetic treatment of the solar wind protons and the heavy cometary ions allows us to examine two important issues. One is the effect of the velocity distribution function of the heavy ions on the shock formation and structure, and the other is the degree of coupling between the two ion species. The result of this study indicate that at high Mach numbers the shock structure is highly dependent upon the velocity distribution of the heavy ions. For example, when the newly created ions comprise a ring distribution in the solar wind frame, most of them turn around downstream of the shock surface and reenter the upstream region to form a large foot that extends about a heavy ions gyroradius upstream of the shock. On the other hand, heavy ions which have been picked up by the solar wind and possesses a Maxwellian distribution can mostly penetrate the shock without returning upstream and affecting the shock structure as much. In either case, however, at high Mach numbers the shock strength is the same. At low Mach numbers, where the shock is weak, the velocity distribution of heavy ions has a smaller effect on the formation of shock and its structure. In this regime, the degree of coupling between the cometary ions and the solar wind protons and the corresponding critical Mach number (at which a shock should begin to form) are determined from a set of Rankine-Hugoniot relations. The results of the simulations suggest that some coupling does occur (evidently, through the electromagnetic fields, since there are no particle collisions in the calculations), but less than that expected from magnetohydrodynamics. For low Mach numbers, it is also shown that shocks have a transitory nature, where they are continuously formed by the protons and subsequently destroyed by the heavy ions.     

Instability of the structure of strong oblique MHD cosmic-ray shocks

A criterion of the instability of a flow of a thermal plasma and cosmic rays in front of an oblique MHD shock wave with respect to short-wavelength magnetosonic disturbances is derived. The dependence of a cosmic-ray diffusion tensor on a plasma density and a large-scale magnetic field is taken into account. The most unstable disturbances propagate at an angle to the magnetic field if diffusion is strongly anisotropic. In some cases the most strong instability connects with the off-diagonal terms of the diffusion tensor.     

A structure and energy dissipation efficiency of relativistic reconfinement shocks

We present a semi-analytical hydrodynamical model for the structure of reconfinement shocks formed in astrophysical relativistic jets interacting with external medium. We take into account exact conservation laws, both across the shock front and in the zone of the shocked matter, and exact angular relations. Our results confirm a good accuracy of the approximate formulae derived by Komissarov & Falle. However, including the transverse pressure gradient in the shocked jet, we predict an absolute size of the shock to be about twice larger. We calculate the efficiency of the kinetic energy dissipation in the shock and show a strong dependence on both the bulk Lorentz factor and opening angle of the jet.     

Ionization structure in accretion shocks with a composite cooling function

We have investigated the ionization structure of the post-shock regions of magnetic cataclysmic variables, using an analytic density and temperature structure model in which effects caused by bremsstrahlung and cyclotron cooling are considered. We find that in the majority of the shock-heated region where H- and He-like lines of the heavy elements are emitted, the collisional-ionization and corona-condition approximations are justified. We have calculated the line emissivity and ionization profiles for iron as a function of height within the post-shock flow. For low-mass white dwarfs, line emission takes place near the shock. For high-mass white dwarfs, most of the line emission takes place in regions well below the shock and hence it is less sensitive to the shock temperature. Thus, the line ratios are useful to determine the white dwarf masses for the low-mass white dwarfs, but the method is less reliable when the white dwarfs are massive. Line spectra can, however, be used to map the hydrodynamic structure of the post-shock accretion flow.     

Circumstellar shocks

The presence of shocks in a wide variety of circumstellar phenomena will be illustrated and discussed.     

The physics of particle acceleration by collisionless shocks

Using analytic theory, test-particle simulations, and self-consistent hybrid simulations, we show that quasi-perpendicular shocks—those which propagate nearly perpendicular to the upstream magnetic field—accelerate particles directly out of the incident thermal population to energies much higher than the upstream ram energy of the plasma. It has already been established that quasi-parallel shocks—those which propagate nearly in the same direction as the upstream magnetic field—efficiently accelerate particles directly out of the incident thermal population; however, this has not yet been established for quasi-perpendicular shocks. Our results can be understood within the framework of the diffusive shock acceleration theory. We find that the accelerated-particle spectrum obtained from a more-general self-consistent hybrid plasma simulation are quantitatively consistent with a less-sophisticated test-particle simulation. The implications of this are discussed.     

The interaction principle in radiative transfer

We describe the interaction principle which is of fundamental importance to the theory of radiative transfer in one-, two-, and three-dimensional geometry. We also describe the practical difficulties associated with this principle in these geometries.     

The radiative relaxation time in the chromosphere

The cooling effect of emission in the spectral lines, which dominates over continuous emission in the chromosphere and becomes important first around the temperature minimum, modifies greatly the radiative relaxation timet _r in the solar atmosphere. This rises from low photospheric values to a maximum of 8 min just aboveT _min, falls in the low chromosphere to 1.5 min because of line emission, but rises again to 6 min atT 7000–8400 K in the chromosphere where hydrogen ionization increases the specific heat.