\section{ INTRODUCTION } 

									
Radiative shock waves arise in a wide range of high-speed astrophysical 
fluid flows. They play important if not the dominant role in the 
emission from high-energy systems 
ranging from compact x-ray binaries to supernovae and supernova remnants
and to young stellar systems, such as classical T Tauri stars and
Herbig-Haro objects. In most if not all of the aformentioneed systems, 
the fluids are likely to be strongly magnetic. In particular, it is  
well-established that magnetic fields dominate the large scale behavior 
and accretion of the high-speed flows onto the accreting objects 
in the neutron star compact x-ray binary systems knowns as the
{\it Massive and Low Mass X-ray Pulsar Systems}, the
white dwarf compact x-ray binary systems known as the
{\it Polars} and {\it Intermediate Polars} (IPs), and in the 
classical T Tauri protostellar systems. The complete utilization 
of the wealth of available observational data which has been 
acquired on these systems by NASA missions
reaching back to the 1970s and which continues to 
be amassed to this day, requires detailed modeling and 
understanding of the steady-state and time-dependent
behavior of radiating shocks in strongly magnetic plasmas.


In magnetic systems, supersonic accretion flows are forced to flow 
along the strong magnetic field lines of the accreting object and thus 
flow roughly radially onto the surface of the star. In order to merge
smoothly onto the star's surface, strong radiating shock waves form
in the flows. 
The stability of radiative shocks with power law cooling 
functions
\begin{equation}
\Lambda = \Lambda_{\circ} \rho^{\beta} T^{\alpha}
\end{equation}
where $\Lambda_{\circ}$, $\beta$, and $\alpha$ are constants, has been 
extensively investigated for nonmagnetic plasma flows (Langer, Chanmugam, 
\& Shaviv 1981,1981a, Chevalier \& Imamura 1982, Imamura, Wolff, \& Durisen 
1984, Imamura 1985, Bertschinger 1986, Wolff, Gardner, \& Wood 1989,
Imamura \& Wolff 1991, Imamura {\it et al}. 1996, Sexton \& Wu 200x). It was
found that radiative shocks are unstable when the front is perturbed normal
to its face for a cooling function with weak temperature
dependence.  The lowest order oscillation mode, the F mode,
is unstable when $\alpha$ $\buildrel<\over\sim$ 0.4  and $\beta$ = 2. 
Higher order oscillation modes are 
more unstable than the F mode and instability in them 
persists to larger $\alpha$. When the perturbation is 
taken to be two-dimensional (2-D) which corresponds to a rippling or 
corrugation of the shock front, radiative shocks are more unstable. 
The F mode for shocks with $\alpha$ = 0.5 and $\beta$ = 2 are 
unstable when 2-D perturbations are allowed. 


The case of magnetic plasma shocks is far less studied
and, consequently, not nearly as well-understood. 
The addition of the magnetic field increases 
the degrees-of-freedom available to the flow which leads to 
a richer spectrum of possible behaviors. To appreciate this, note that
for a strong shock in a nonmagnetic plasma, a corrugation of the shock front
generates pressure waves but that such waves are unable to propagate upstream
because the preshock flow speed is greater than the sound speed
$|v|$ $>$ $c_s$ so that the waves are dragged toward the obstacle faster
than they can propagate away. In a magnetic plasma, if we have the 
ordering $v_A$ $>$ $|v|$ $>$ $c_s$, where $v_A$ is the Alf\'en speed, 
a hydrodynamic shock can still form
but, in this case, magnetohydrodynamic waves generated by 
the corrugation of the front may be able to propagate upstream ahead 
of the shock. The stability of a magnetic radiative shock when the 
shock front is rippled then involves a determination of the
thermal stability of the emission region taking into 
account the generation of magnetohydrodynamic waves 
which alter the preshock plasma and 
may carry energy and momentum through the cooling region. 
The stability of magnetic shocks was investigated for magnetic shocks
dominated by bremsstrahlung by Edelman (1991a,1991b). Here, 
we further investigate the stability properties of shocks in 
strongly magnetic plasmas.


\section{ PHYSICAL PICTURE }


Radiative shocks are composed of four regions: the preshock flow,
a narrow shock transition where the plasma heats and compresses, 
the postshock cooling region where the shock heated plasma radiates
away its internal energy and decelerates, and a cold layer at the 
base of the flow where the accreting plasma merges with the 
driving piston (star). See Figure 1 for a schematic diagram of 
a radiative shock wave.
\begin{figure}
\includegraphics[width=6.0in]{fig_1_radiative_shock.eps}
\caption{
Radiative Shock Structure. Radiative shock waves are composed of four 
regions: the preshock flow (region 1); the shock transition; the 
postshock cooling region (the emission region, region 2); and the cold 
driving layer (the star or the driving pistion, region 3). The shock
transition is taken to be a discontnuous change in the flow properties.
}
\end{figure}
We model parallel, strong, radiative shocks in the ideal MHD approximation.
By parallel, we mean shocks where the preshock flow and the magnetic field
are perpendicular to the shock front. By strong, we mean shocks where the 
preshock flow velocity is large compared to the sound speed, that is, 
$|v|/c_s$ $\gg$ 1. Here, the isothermal sound speed is given by 
$c_s$ = $\sqrt{\gamma P/\rho}$, $\gamma$ is the adiabatic index, 
$P$ is gas pressure and $\rho$ is mass density. 
In ideal MHD, the conductivity, $\sigma$
is taken to be infinite so that the resistivity $\eta$ =  0. We expect
several types of shocks to form in MHD, shock types corresponding to 
the different wave modes supported by adiabatic, magnetic plasmas.
One type corresponds to Alfv\'en modes, 
two correspond to the slow and fast magnetosonic modes, and a 
final type which corresponds to entropy waves. 
Alfv\'en waves are characterized by disturbances
perpendicular to the plane defined by the wave propagation
vector and the equilibrium magnetic field. Entropy waves have zero
frequency whose disturbances are advected with the flow. 
Magneotsonic modes are
chararcterized by disturbances in the plane defined by the wave 
propagation vector and the equilibrium magnetic field.  
Magnetosonic waves propagate at speeds given by 
\begin{equation}
\left(\frac{\omega}{k}\right)^2 = 
0.5\left[(c_s^2+v_A^2)
\pm\sqrt{(c_s^2+v_A^2)^2-4v_A^2c_s^2{\rm cos}^2\psi}\right]
\end{equation}
where $\omega$ is the frequency, $k$ is the wave vector, $v_A$ is the
Alfv\'en velocity, $c_s$ is the sound speed, cos $\psi$ = ${\bf k}\cdot
{\bf B}_{\circ}/
\|{\bf k}\|\|{\bf B_{\circ}}\|$,
and {\bf B}$_{\circ}$ is the equilibrium magnetic field. The upper sign for
the radical corresponds to the fast magnetosonic mode while the lower sign
corresponds to the slow magnetosonic mode. Consider the slow mode first.
For cos $\psi$ = 0, $\omega$ = 0 and there is no slow mode. For 
cos $\psi$ = 1, the wave speed is $c_s$ or $v_A$, the Alfv\'en speed,
whichever is larger. For the fast 
mode, the minimum wave speed is $v_A$ and the 
maximum wave speed is $\sqrt{v_A^2+c_s^2}$.
Alfv\'en waves propagate at $v_A$. Magnetosonic
waves and Alfv\'en waves may propagate upstream
as well as downstream of MHD shock waves for sub-Alfv\'enic flows, flows with
$v_A$ $>$ $|v|$.


The physical picture is then that of a supersonic, fully ionized,
adiabatic plasma flowing along magnetic 
field lines toward some obstacle, for exmaple, a star 
(see Figure 2). At a small distance $h$ from the obstacle, a shock forms
and the plasma heats to a temperature on the order of
\begin{equation}
kT_s = \frac{1}{3} m v_{in}^2
\end{equation}
where $k$ is the Boltzmann constant, $T_s$ is the postshock temperature, 
$m$ is the average mass of the incoming particles and $v_{in}$ is 
the bulk speed of the incoming  flow. The shock
heated plasma then cools via radiation as it 
settles onto and merges with the obstacle. The
characteristic size of the cooling region is 
\begin{equation}
h = \frac{1}{4}\;v_{in}\tau_c
\end{equation}
where $\frac{1}{4}v_{in}$ is the postshock velocity, 
$\tau_c$ is the characteristic postshock cooling time scale 
\begin{equation}
\tau_c = \frac {3}{2} \left(\frac {n_skT_s} {\Lambda_s}\right),
\end{equation}
$n_s$ = 4 $n_{in}$ is the number density of the postshock plasma, 
$n_{in}$ is the number density of the preshock flow, and $\Lambda_s$ 
is the cooling function evaluated at the shock. The size 
of the emission region scales with the shock parameters as 
\begin{equation}
h \propto n_s^{1-\beta}T_s^{1-\alpha}.
\end{equation}

\begin{figure}
\includegraphics[width=4.0in,angle=270]{wd_dipole.ps}
\caption{
The plasma is loaded onto the field
lines near the inner Lagrangian point of the system
in Polars and near the inner edge of an accretion disk in IPs.
For both cases, this occurs in or near the orbital plane. The
plasma then flows toward the white dwarf following the magnetic field 
lines. A radiating shock wave forms as the plasma settles onto the 
surface of the white dwarf.
For white dwarf systems, the importance of the magnetic field is
measured by the parameter
$
\left(v_A/v_{in}\right)^2
= 230 \left( L_*/L_{\odot} \right)^{-1}
\left( f/10^{-4} \right)
\left( M_*/M_{\odot} \right)^{1/2}
\left( B_*/10\;MG \right)^2
\left( R_*/5\times10^{8}\;cm \right)^{3/2}
$
where  $v_A$ is the Alfv\'en speed, $v_{in}$ is the preshock flow velocity,
$L_*$ is the system luminosity, $L_{\odot}$ is the Solar luminosity, 
$f$ is the fraction of the stellar surface covered by the accretion funnel,
$M_*$ is the white dwarf mass, $M_{\odot}$ is the Solar mass, $B_*$ is
the magnetic field of the white dwarf, $MG$ stands for MegaGauss,  and 
$R_*$ is the white dwarf radius. $f$ $\sim$ $10^{-4}$ for Polars and
$f$ $\sim$ $10^{-2}$ for IPs.
}
\end{figure}



Why shocks for $\Lambda$ with 
strong temperature dependence are stable while ones
for $\Lambda$ with weak temperature dependence are 
unstable is easily understood from the form for $h$. 
First, perturb the shock outward so that the relative
speed between the shock front and the incoming plasma increases. 
This causes the postshock temperature
$T_s$ to increase. If $\alpha <$ 1, the emission region thickness $h$ 
increases in size and the plasma is unable to cool by the time
it reaches the obstacle. The pressure in the emission region 
then increases. This forces the shock to continue
moving outward and the emission region is unstable. 
For $\alpha >$ 1, $h$ decreases because $T_s$ 
increases as the shock moves outward. That is,
the plasma cools more efficiently and the emission region loses
pressure support. The outward motion then slows and the shock front
soon starts to fall back on itself. Radiative shocks are expected to
be stable for large $\alpha$. This dimensional argument is verified by
detailed calculations although the critical $\alpha$ turns out to be 
somewhat smaller; the critical value is $\alpha_c$ = 0.4 for the lowest 
order oscillation mode, the F mode (Chevalier \& Imamura 1982).


The addition of a magnetic field alters the picture in that 
MHD waves may carry energy away from the shock modifying the 
incoming flow and acting as an additional flux
term. The determination of the thermal stability of the
postshock cooling region requires consideration of how
the MHD waves interact with the preshock flow and 
postshock cooling plasma. It are these interactions
we include and investigate here.


\section{ NUMERICAL MODEL }


\subsection{ Magnetohydrodynamic (MHD) Equations }


The ideal MHD equations are
\begin{equation}
\left( \frac {\partial} {\partial t} + {\bf v \cdot \nabla} \right) \rho 
+ \rho ( {\bf \nabla \cdot v} ) = 0,
\end{equation}
\begin{equation}
\rho \left( \frac {\partial} {\partial t} + {\bf v\cdot\nabla} \right) {\bf v}
= - {\bf \nabla} P + ({\bf j \times B}) / c,
\end{equation}
\begin{equation}
\rho T \left( \frac {\partial} {\partial t}+{\bf v}\cdot\nabla \right) S
= (\Gamma-\Lambda)
\end{equation}
and 
\begin{equation}
\frac {\partial}{\partial t}{\bf B} = {\bf \nabla\times}
\left({\bf v}\times{\bf B}
\right)
\end{equation}
Here, $t$ is the time, {\bf v} is the velocity, $\rho$ is the mass              
density, $P$ is the pressure, $T$ is the temperature, $S$ is the entropy
for an ideal gas given by $S$ = $c_V$ln$\;(P\rho^{-\gamma})$, $c_V$ is
the specific heat capacity at constant volume,
$\Gamma$ and $\Lambda$ are the local volume heating and cooling rates,
{\bf B} is the magnetic         
field strength, $\gamma$ is the adiabatic index, 
and ${\bf j}$, 
the current density, is given by
\begin{equation}                                                        
{\bf j} =   \frac {c}{4\pi} {\bf \nabla \times B}
\end{equation}
We use Cartesian coordinates and assume          
that the unperturbed flow and magnetic field are along the z-axis.  

\clearpage

\begin{thebibliography}{}

\bibitem[\protect\citename{Aizu 1973}]{Az73}
Aizu, K. 1973, Progr Theor Phys, 49, 1184 

\bibitem[\protect\citename{Bertschinger 1973}]{Ber73}
Bertschinger, E. 1986, {\it ApJ}, 304, 154

\bibitem[\protect\citename{Chevalier \& Imamura 1982}]{CI82}
Chevalier, R. A. \& Imamura, J. N. 1982, {\it ApJ}, 261, 543

\bibitem[\protect\citename{Dgani \& Soker 1994}]{DS94}
Dgani, R. \& Soker, N. 1994, {\it ApJ}, 434, 262

\bibitem[\protect\citename{Draine \& McKee 1993}]{DM93}
Draine, B. \& McKee, C. F. 1993, {\it ARAA}, 31, 373

\bibitem[\protect\citename{Edelman 1990a}]{Ed90a}
Edelman, M. 1990a, {\it Astrophysics}, 31, 656

\bibitem[\protect\citename{Edelman 1990b}]{Ed90b}
Edelman, M. 1990b, {\it Astrophysics}, 31, 758

\bibitem[\protect\citename{Imamura 1985}]{Im85}
Imamura, J. N. 1985, {\it ApJ}, 296, 128

\bibitem[\protect\citename{Imamura, Wolff \& Durisen 1984}]{IWD84}
Imamura, J. N., Wolff, M. T., \& Durisen, R. H. 1984, {\it ApJ}, 276,
667

\bibitem[\protect\citename{Langer, Chanmugam, \& Shaviv 1981}]{LCS81}
Langer, S. H., Chanmugam, G., \& Shaviv, G. 1981, {\it ApJ}, 245, 223

\bibitem[\protect\citename{Toth \& Draine 1993}]{TD93}
Toth, G. \& Draine, B. L. 1993, {\it ApJ}, 4113, 176

\bibitem[\protect\citename{Wolff, Gardner, \& Wood 1989}]{WGW89}
Wolff, M. T., Gardner, J., \& Wood, K. S., 1989, {\it ApJ}, 346, 833

\end{thebibliography}