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Calculation of the radiative shock structure

When the shock wave velocity increases, the internal degrees of freedom of atoms are more and more excited and consequently, a radiative flux is produced in the recombination/de-excitation region of the shock wake. Because the gas between this region and the shock front is optically thin and the shock velocity is quite smaller than the light velocity, a large part of the incoming flux is absorbed by the unperturbed cooler gas located ahead the front of the shock (Fig.[*]). If the UV-flux is large enough, many photoexcitations and photoionizations occur in a precursor region and consequently, the electronic number density ne substantially increases at the shock front. Thus, because collisional ionizations become more efficient in the shock wake, the intensity of the radiative flux increases. Finally, it appears that if the flux is appreciable, the structures of the both precursor and wake are coupled and only a self-consistent approach can provide the correct solution. Such a simple and natural phenomenon dramatically complicates the calculation of the radiative shock structure because the hydrodynamical equations are coupled with transfer equation which is nonlocal and strongly depends on the spectral frequency.


  
Figure: The principle of the radiative coupling in a shock wave.
\begin{figure}
\epsfxsize=90mm
\epsfbox{figure2.ps}\end{figure}

A direct consequence of the presence of a perceptible radiative field induced by the shock passage, is a strong compression which can reach several hundreds. Indeed, it is well known that the viscous shock front compresses the gas up to the limiting ratio $(\gamma+1)/(\gamma-1)$ where $\gamma$ is the specific heat ratio. Thus, we must expect a compression not larger than 4 or 6 if the gas is atomic or diatomic, respectively. When the shock intensity increases, then the internal degrees of freedom of molecules are excited and a radiative field appears in the shock wave layers. This phenomenon leads to an additional compression which is considerable when the shock becomes nearly isothermal, i.e. when the emission of the radiative energy by the shock wave is extremely rapid in comparison with hydrodynamical relaxation processes. In the limit compression ratios proportional to M2 are expected. In these conditions of hypersonic shocks, the radiative field has a major effect on the shocked gas.

Moreover, such as large compression effect leads to a substantial amplification of the turbulence level of the unperturbed gas. Indeed, as shown by current theories about the interaction between a shock wave and a free turbulence (see for instance [6]), the amplification depends on the density compression rate. Thus, the calculation of the detailled structure of radiative shocks represents a required step in the study of the turbulence amplification.

Because the calculation of the structure of a radiative shock wave, including both the precursor and the wake, is a very complicated problem, only the one-dimensional case has been considered up to now. It is assumed that the shock energy is large enough to restrict the problem to the steady case. No transport phenomena such as viscosity, conductivity and diffusion are considered. The atomic viscosity is only relevant inside the shock front but its structure is not calculated here. Although, the effect of the electronic conductivity in the precursor region can be appreciable ([13], [14]), it is rarely included in the shock models. Finally, convection, turbulence or magnetic field are not considered. Due to the relatively large density and low temperature in stellar atmospheres, the magnetic dissipation becomes only important if the magnetic field is extremely strong which is not typical.


  
Figure: Small part of the absorption spectrum for the solar abundance for a gas at T=8000K and $\rho=10^{-8}$gcm-3 computed with data from TOPbase. The full spectrum is based on about 2 millions of bound-bound transitions.
\begin{figure}
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\epsfbox{figure3.ps}\end{figure}

Three kinds of equations provide the shock structure. The first ones are the chemical equations. The mass conservations of species i is given by  
 \begin{displaymath}
\frac{\partial\rho_{i}}{\partial\,t}+\frac{\partial(\rho_{i}u)}{\partial\,x}
=w_{i}\end{displaymath} (1)

where the mass density is $\rho_{i}=m_{i}n_{i}$, mi is the mass of the species i and ni its number density; wi is the net rate of production of species i which depends on the different chemical reactions (creation and destruction) occurring in the gas. Due to the steady assumption, the time derivative can be omitted.

Then, we have the three hydrodynamical conservation equations:

Because in the thermalization region just behind the shock front, the electronic temperature is not equal to that of heavy particles (i.e. atoms and molecules), an additional conservation equation for the electronic energy is required:  
 \begin{displaymath}
u\frac{de_{e}}{dx}+up_{e}\frac{d(1/\rho)}{dx}=q_{el}+q_{in}\,,\end{displaymath} (5)

where ee is the translational energy of electrons per unit mass, pe their pressure and qel and qin stand for the energy gain of electrons by the elastic and inelastic collisions with ions and atoms respectively.

Thus, in these equations, the radiative field only occurs through three different terms pr, er and Fr. In stellar atmospheres, in which the shock Mach number rarely exceeds 30, only the radiative flux Fr must be considered. This results from the fact that  
 \begin{displaymath}
\frac{F_{r}/\rho\,u}{e_{r}}\simeq\frac{F_{r}/\rho\,u}{p_{r}/\rho}
\simeq\frac{c}{u}\gg1\end{displaymath} (6)

and that the radiative energy and pressure terms become of the order of the gas energy and pressure for extremely fast shocks such as in supernovae explosions where the initial shock front velocity is around 5% of the speed of light. Consequently, both radiative energy and pressure terms are usually neglected in stellar atmosphere. Finally, the last equation is the radiative transfer equation  
 \begin{displaymath}
\mu\frac{dI(\mu,x,\nu)}{dx}=j(x,\nu)-\kappa(x,\nu)I(\mu,x,\nu)\end{displaymath} (7)

where I is the specific intensity, $\mu\equiv\cos\theta$ the directional cosine and j and $\kappa$ the emission and absorption coefficients respectively.


next up previous
Next: The ``mean-photon'' approximation Up: On the Structure of Previous: Radiative shocks in stellar

9/11/1998