2 Two-fluid radiation hydrodynamics

2.1 General approach

We solve the stationary, one-dimensional, two-fluid hydrodynamic equations simultaneous with the frequency and angle-dependent radiative transfer, closely following the approach of WB96. We deviate from WB96 in the treatment of the shock itself. Instead of integrating the flow through the shock with an artificial viscosity, we adopt the presence of a strong ion shock and start the integration with adopted values of the ion and electron temperatures (see below). Of course, the solution now fails to reproduce the rapid rise in ion temperature across the shock, but otherwise the results are practically identical except for small differences at low velocities where the flow connects to the atmosphere of the star and large gradients revive the viscous terms again. The set of differential equations then reads (compare Eqs. (1) to (4), (7), and (8) of WB96)

$$\rho \upsilon = - \dot m$$ (1)

$$\rho \upsilon \,\frac{{\rm d}\upsilon}{{\rm d}x} + \frac{{\rm d}}{{\rm d}x}(P_{\rm i} + P_{\rm e}) = - g$$ (2)

$$\rho \upsilon \,\frac{{\rm d}{\cal E_{\rm i}}}{{\rm d}x} - \u... ... E_{\rm i}}) \frac{{\rm d}\rho}{{\rm d}x} = - \Lambda_{\rm ei}$$ (3)

$$\rho \upsilon \,\frac{{\rm d}{\cal E_{\rm e}}}{{\rm d}x} - \u... ...= \Lambda_{\rm ei} -\rho\, \frac{{\rm d}F_{\rm rad}}{{\rm d}x}$$ (4)

$$\rho\,{\rm cos}\vartheta\frac{{\rm d}I_{\nu}(\vartheta)}{{\rm... ... - \kappa_{\nu}(\vartheta) B_{\nu}(T_{\rm e}) - \sigma J_{\nu}$$ (5)

$$F_{\rm rad} = 2\pi\,\int\limits^\infty_0\,\int\limits^{+1}_{-... ...rtheta)\,{\rm cos}\vartheta\,{\rm d(cos\vartheta)}\,{\rm d}\nu$$ (6)

where $\rho$ is the mass density, $\upsilon$ the velocity, $\dot m$the mass flow density, $P_{\rm i}$ and $P_{\rm e}$ the ion and electron pressures, ${\cal E_{\rm i}}$ and ${\cal E_{\rm e}}$ ion and electron internal energy densities, $I_{\nu}(\vartheta)$ the specific intensity of the radiation field at frequency $\nu$ and angle $\vartheta$, $J_{\nu}$ the angle-averaged intensity, $\kappa_{\nu}(\vartheta)$ the angle-dependent absorption coefficient, $\sigma$ the Thomson scattering coefficient, $B_{\nu}(T_{\rm e})$ the Planck function at electron temperature $T_{\rm e}$, and $F_{\rm rad}$the total radial energy flux in the radiation field. We choose the downstream column density x as the independent variable rather than the radial coordinate or the geometrical height h. Following WB96, we neglect the effects of radiation pressure and thermal conduction.

The connecting link between the hydrodynamics (Eqs. (1) to (4)) and the radiative transfer (Eqs. (5) and (6)) is $F_{\rm rad}$: the electron gas cools by radiation and is heated by Coulomb interactions with the ions, described by the non-relativistic electron ion energy exchange rate $\Lambda_{\rm ei}$ (Spitzer 1956, see also WB96, their Eqs. (5) and (6)). The fully angle and frequency-dependent radiative transfer accounts for cyclotron absorption, free-free absorption, and coherent electron scattering. Our emphasis is on the largely correct treatment of the cyclotron spectra and we accept inaccuracies of the hard X-ray spectra caused by the neglect of Compton scattering. This still rather general treatment ensures that our results are relevant for a wide range of $\dot m$ including the low-$\dot m$ regime where radiative losses by optically thick cyclotron radiation dominate.The cyclotron absorption coefficients used here are the added coefficients for the ordinary and the extraordinary rays (WB92). This limitation is dropped in Sect. 3.1, below.

We use a Rybicki code for the LTE radiative transfer and integrate the set of equations implicitly, using a Newton scheme to iterate between hydrodynamics and radiation transport. For more details see WB96. Our solution is strictly valid only for an infinite plane parallel layer. A first-order correction to the peak electron temperature for emission regions of finite lateral extent D (Fig. 1) is discussed in Sects. 3.2 and 3.3 below.

Equation (2) accounts for post-shock acceleration and heating of the flow by the constant gravity term $g =\,$G $M_{\rm wd}$/ $R_{\rm wd}$². Within our one-dimensional approximation which disregards the convergence of the polar field lines, considering the variation of gravity with radius would not be appropriate. Our approach is, therefore, limited to stand-off distances of the shock $h_{\rm sh}$$\ll$ $R_{\rm wd}$. Settling solutions with $h_{\rm sh}$ $R_{\rm wd}$ are not considered.

As in WB96, we assume that the pre-shock flow is fully ionized, but cold. Soft X-rays will photoionize the infalling matter and create a Strömgren region with a temperature typical of planetary nebulae, but for our purposes this is cold. Heating of the pre-shock electrons by thermal conduction may be more important. Equilibrium between diffusion and convection defines an electron precursor with a radial extent  $\lambda_{\rm pre} \simeq 4\times 10^{-15}\ T_{\rm e,s}^{5/2}/\dot m$ cm, where $T_{\rm e,s}$ is the electron temperature at the shock in K (Imamura et al. 1987) and $\dot m$ is in gcm^-2s^-1. Near the one-fluid limit, electron and ion shock temperatures are similar, $T_{\rm e,s} \simeq T_{\rm i,s}$, and the precursor extends to $\lambda_{\rm pre} \simeq 0.09\,$ $h_{\rm sh}$. In a cyclotron-dominated plane-parallel flow, however, two effects cause the precursor to be less important: (i) the electrons never reach the peak temperature expected from one-fluid theory and (ii) the optically thick radiative transfer in the plane-parallel geometry sets up a radial temperature gradient which further depresses the electron temperature at the shock. In this paper, we do not consider thermal conduction, neglect the presence of the electron precursor, and opt to set $T_{\rm e,s}$= 0.

At x = 0, we adopt the Rankine-Hugoniot jump conditions for a gas with adiabatic index 5/3, i.e. we set the post-shock density to 4 $\rho_{\rm o}$, the bulk velocity to $\upsilon _{\rm o}$/4, and the pressure to (3/4) $\rho_{\rm o}$ $\upsilon _{\rm o}$², with $\rho_{\rm o}$ and $\upsilon _{\rm o}$ the density and bulk velocity in the pre-shock flow. With $T_{\rm e,s}$= 0, the ion shock temperature is

$$T_{\rm i,s} = 3\,\mu_{\rm i}m_{\rm u}/(16\,k)\,\upsilon_{\rm o}^2.$$ (7)

We use $\upsilon _{\rm o}$=(2G $M_{\rm wd}$/ $R_{\rm wd}$)^1/2 with Nauenberg's (1972) relation between mass and radius of the white dwarf. $\mu_{\rm i}$ is the molecular weight of the ions, ${m}_{\rm u}$ the mass unit, and k the Boltzmann constant.

All numerical calculations are performed for a hydrogen plasma with $\mu = 0.5$ and $\mu_{\rm i} = \mu_{\rm e} = \mu_{Z} =1$, where $\mu_{\rm e}$ is the number of nucleons per electron, $\mu$ is the molecular weight of all particles, and $\mu_{Z}$ the molecular weight of the ions weighted with $Z_{\rm k}^2$. We include the molecular weight dependence in our equations in order to allow conversion to other compositions, e.g., a fully ionized plasma of solar composition with $\mu = 0.617$, $\mu_{\rm i} = 1.297$, $\mu_{\rm e} = 1.176$, and $\mu_{Z} = 0.927$.

  
2.2 Bremsstrahlung and cyclotron emissivities

The frequency-integrated volume emissivity for brems- strahlung is

$$\epsilon_{\rm brems} = {c}_2 T^{1/2} n_{\rm e} \sum\limits_{... ...1/2}} \frac{\mu^{1/2}}{\mu_{\rm e} \mu_{Z}}\,P^{1/2}\rho^{3/2}$$ (8)

where $c_2 = 1.43\times 10^{-27}$ cgs-units, $n_{\rm e}$ is the number density of the electrons, n_k the number density of the ions of charge Z_ke, P is the gas pressure, and $\rho$the mass density. The Thomson scattering optical depth of a bremsstrahlung dominated flow parallel to the flow is of order unity, implying that bremsstrahlung is essentially optically thin (Aizu 1973). We use the cyclotron absorption coefficients for the ordinary and extraordinary rays given by Chanmugam & Dulk (1981), Thompson & Cawthorne (1987), and WB92. The total cyclotron emissivity of non-relativistic electrons integrated over wavelength and solid angle is

$$\epsilon_{\rm cyc} \propto n_{\rm e}\,T\,B^2 \propto (\mu/\mu_{\rm e})\,B^2 P.$$ (9)

In the columns considered, cyclotron radiation is optically thin in the higher harmonics, but is always optically thick in the first few harmonics, and the temperature distribution in a cyclotron-dominated emission region can properly be calculated only by solving the coupled radiation-hydrodynamic equations.

Figure 1: Schematics of the emission region. The region is bounded at the top by the shock front and at the bottom by the white dwarf.

2.3 Geometry of the emission region

Figure 1 shows the schematic of an emission region with finite lateral extent D. The shock is located at $h =\,$ $h_{\rm sh}$ above the white dwarf surface. The downstream column density is x = 0 at the shock and and $x\,=\,$$x_{\rm s}$ at the surface of the star, with x and hbeing related by d$x = -\rho$dh. The gravity vector $\vec{g}$ and the magnetic field vector $\vec{B}$ are taken parallel to the flow lines. The radiation-hydrodynamic equations are solved for layers of infinite D to yield the run of electron temperature and mass density, $T_{\rm e}(x)$ and $\rho(x)$. These profiles are later employed to calculate the outgoing spectra for emission regions with finite D by ray tracing, i.e. by adding the contributions from an appropriate number of rays (Fig. 1 and Sect. 3.1). This procedure is not self-consistent if optically thick radiative losses occur from the sides of the column. An appropriate first-order correction to the temperature structure derived for the infinite layer is discussed in Sects. 3.2 and 3.3, below. The treatment of really tall columns requires a different approach which specifically allows for the emission from the sides of the column (Wu et al. 1994).

Radiation intercepted by the white dwarf is either reflected or absorbed and reemitted by its locally heated atmosphere. We assume coherent scattering of hard X-rays using the frequency-dependent reflection albedo $A_{\nu}$ of van Teeseling et al. (1994). The fraction $1 - A_{\nu}$ of the energy is re-emitted in the UV and soft X-ray regime and is not considered in this paper.

2.4 Limiting cases

Here, we consider simple limiting cases which can, in part, be solved analytically. Below, we shall discuss our numerical results in terms of these limiting solutions. The high $\dot m$, low B limit is the bremsstrahlung-dominated one-fluid solution. In the opposite limit of low $\dot m$, high B one enters the non-hydrodynamic regime (Lamb & Masters 1977). Here, the bombardment solution of a static atmosphere heated by a stream of fast ions and cooling by cyclotron emission is an appropriate approximation (Kuijpers & Pringle 1982, WB92, WB93).

The one-dimensional, one-fluid hydrodynamic equations with simple terms for optically thin cooling can be solved analytically (Aizu 1973; Chevalier & Imamura 1982). Integration of Eq. (2) with Eq. (1), $P = P_{\rm i} + P_{\rm e}$, and g = 0 yields $P = \dot m(\upsilon_{\rm o} - \upsilon)$ which allows to express the emissivities of Sect. 2.2 as $\epsilon_{\rm brems} \propto \dot m^2~{f}(\upsilon)$ and $\epsilon_{\rm cyc} \propto \dot m~{\rm g}(\upsilon)$, with f and g being functions of the flow velocity $\upsilon$ and with additional dependencies on the $\mu$'s and B contained in the proportionality factors. Integration of the energy equation over $\upsilon$ yields expressions for the column density $x_{\rm s}$ and the geometrical shock height $h_{\rm sh}$ which reflect the parameter dependence of $\epsilon$,

  

$$x_{\rm s} \propto \frac{\dot m^2}{\epsilon}\hspace{2mm} \left\{ \begin{array}{@{\hs... ...&\propto & \dot m &{\rm for}& \epsilon = \epsilon_{\rm cyc}, \end{array}\right.$$ (10)

$$h_{\rm sh} \propto \frac{\dot m}{\epsilon}\hspace{2mm} \left\{ \begin{array}{@{\hspa... ...o &{\rm constant} &{\rm for}& \epsilon = \epsilon_{\rm cyc}. \end{array}\right.$$ (11)

In the flows considered here, bremsstrahlung is close to optically thin and the analytical solution is quantitatively corroborated by our numerical results. Cyclotron emission, on the other hand, is optically thick in the lower harmonics which reduces the effective emissivity and inflates the emission region. While the $\dot m$-dependence of cyclotron-dominated columns in Eqs. (10) and (11) is recovered in our numerical calculations, not surprisingly, the numerically derived sizes of $x_{\rm s}$ and $h_{\rm sh}$ are much larger than those predicted by the (unquoted) proportionality factors for cyclotron cooling in Eqs. (10) and (11).

Figure 2: Temperature profiles for the ions (dashed curves) and electrons (solid curves) as functions of column density x for $M_{\rm wd}$= 0.6 $M_{\odot }$ and field strengths of 10MG (left panel), 30MG (center panel) and 100MG (right panel). The individual curves are for mass flow densities $\dot m = 100, 10, 1, 10^{-1}$ and $10^{-2}\,$gcm-2s-1 (from top). In the left panel, the curves for 100 and 10gcm-2s-1 are indistinguishable. In the right panel, the bottom curve is for $10^{-1}\,$gcm-2s-1. From Eqs. (13) and (16), $x_{\rm s, brems} = 0.783$gcm-2 (log $x_{\rm s, brems} = -0.106$) and $T_{\rm max,brems}$ = $T_{\rm i,s}$/2.

  
2.5 Bremsstrahlung-dominated shock solution

For a strong shock in a one-fluid plasma with adiabatic index 5/3, the normalized post-shock velocity $\omega =4\upsilon/$ $\upsilon _{\rm o}$ varies between 1 and 0. The column density x measured from the shock is related to $\omega$ by (Aizu 1973; Chevalier & Imamura 1982)

$$x=c_1\upsilon_{\rm o}^2\left[\sqrt{3}{-}\frac{\pi}{3}{-}\frac... ...2} {+}{\rm cos}^{-1}\left(1{-}\frac{\omega }{2}\right)\right].$$ (12)

The total column density and shock height are given by

  

$$x_{\rm s, brems} c_1\,\upsilon_{\rm o}^2\,(\sqrt{3}-\pi/3) ~~~{\rm g\,cm}^{-2}$$ = (13)

$$h_{\rm sh, brems} c_1\,\upsilon_{\rm o}^3\,\left(39\sqrt{3}-20\pi\right)/ (48\,\dot m)~~~{\rm cm},$$ = (14)

where $c_1 = ({k}^{1/2}{m}_{\rm u}^{3/2}/4\,c_2) (\mu_{\rm e}\,\mu_{Z}/\mu^{1/2})$ with values of $6.22\times10^{-18}$cgs for pure hydrogen and $6.10\times10^{-18}$ cgs for solar composition. The temperature profile follows from pressure equilibrium $P = \rho \upsilon (\upsilon_{\rm o} - \upsilon$), the equation of state for the ideal gas, and Eq. (1) as

$$T = \frac{1}{3}\left(4\omega - \omega ^2\right)T_{\rm max,brems}$$ (15)

with

$$T_{\rm max,brems} = 3\,\mu m{_{\rm u}}/(16\,{k})\,\upsilon_{\rm o}^2 = (\mu/\mu_{\rm i})\,T_{\rm i,s}.$$ (16)

Our two-fluid calculations for high $\dot m$, low B reproduce the temperature profile T(x) given by Eq. (15) with (12) and (16) to better than 1% of $T_{\rm max,brems}$, except for the initial equilibration layer which is infinitely thin in the analytic calculation and has a finite thickness with rising electron temperature in our calculations. As an aside, we note that T/ $T_{\rm max,brems}$ $\,\simeq (1-x/$$x_{\rm s}$)^0.59 with an rms error of less than 1%.

  
2.6 Cyclotron-dominated bombardment solution

The bombardment solution involves by nature a two-fluid approach. WB92 solved this case using a Fokker-Planck formalism to calculate the stopping length of the ions and a Feautrier code for the radiative transfer. WB93 (their Eqs. (8), (9)) provided power law fits to their numerical results for the column density and the peak electron temperature. Since the ions are slowed down by collisions with atmospheric electrons, a factor $\mu_{\rm e}$ appears in $x_{\rm s}$:

 

$$x_{\rm s, bomb} = 3.94\times 10^{-2}\,\mu_{\rm e} (\dot m B_7^{-2.6})^{0.30} M_{\rm wd}^{1.72}~~~{\rm g\,cm}^{-2}$$ (17)

 

$$T_{\rm max,bomb} = 1.28\times 10^9\,(\dot m B_7^{-2.6})^{0.42} M_{\rm wd}^{0.66}~~~~~{\rm K}.$$ (18)

Here, $\dot m$ is in gcm^-2s^-1, B₇ is in units of 10⁷G, and $M_{\rm wd}$ is in solar masses. These fits are very close to the quasi-analytical expressions of Eqs. (5) and (6) of WB93.

With increasing $\dot m$, a shock develops which is initially cyclotron-dominated and ultimately bremsstrahlung-dominated. Since $T_{\rm max,bomb}$ reaches $T_{\rm max,brems}$ at some intermediate $\dot m$, we expect a smooth transition in peak temperature between these cases. The situation is quite different for $x_{\rm s}$, however. At the $\dot m$where $T_{\rm max,bomb}$ equals $T_{\rm max,brems}$, $x_{\rm s,bomb}$ and $x_{\rm s,brems}$ differ by more than two orders of magnitude. The run of $x_{\rm s}$($\dot m$) between these two limiting cases can be determined only with a radiation-hydrodynamical approach.

The bombardment solution does not predict the geometrical scale height of the heated atmosphere which we expect to lie between that of a corona with an external pressure P = 0 and that of a layer compressed by the ram pressure $P =\,$ $\rho_{\rm o}$ $\upsilon _{\rm o}$².

Figure 3: Normalized electron temperature distributions for $M_{\rm wd}$= 0.6 $M_{\odot }$ and the values of B and $\dot m$ given in the figure.

2.7 Parameterization of the results

In the bombardment solution (Eqs. (17), (18)), the dependence of $x_{\rm s}$ and $T_{\max}$ on ${\dot m}B^{-2.6}$ is obtained from the equilibrium between the energy gain by accretion, $F_{\rm acc} \propto \dot m$, and the energy loss by optically thick cyclotron radiation, $F_{\rm cyc} \propto T_{\rm max}\,\omega_*^3$, where $\omega_* = m_*\,\omega_{\rm c}$ is the high-frequency cutoff of the cyclotron spectrum and $\omega_{\rm c} \propto B$ the cyclotron frequency. We determine the limiting harmonic number m_* from the cyclotron calculations of Chanmugam & Langer (1991; their Fig. 5) as $m_* = 4.43\,\Lambda_4^a\,T_8^b$ with $a \simeq 0.12$ and $b \simeq 0.40$. This approximation is valid near depth parameters $\Lambda_4 =\Lambda/10^4 \simeq 1$ and temperatures $T_8 = T_{\rm max}/(10^8\,{\rm K}) \simeq 1$ and is more adequate for the cyclotron-dominated emission regions on polars than the frequently quoted formula of Wada et al. (1980). Replacing $\omega_*$ with $\Lambda \propto x_{\rm s}/B$ in $F_{\rm cyc}$ and equating the accretion and radiative energy fluxes yields the result that a power of $T_{\max}$ is proportional to $\dot m\,B^{3(a-1)} \simeq ~$ ${\dot m}B^{-2.6}$. The same holds for $x_{\rm s}$.

We find that the cyclotron-dominated shocks at low $\dot m$ behave similarly to bombarded atmospheres in that their thermal properties, too, depend on ${\dot m}B^{-2.6}$. The individual temperature profiles $T_{\rm e}(x)$ for different $\dot m, B$ with the same ${\dot m}B^{-2.6}$ coincide only in an approximate way, but the dependency on ${\dot m}B^{-2.6}$ holds quite well for the two characteristic values of each profile, $T_{\max}$ and $x_{\rm s}$. If we leave the exponent $\alpha$ in $\dot m\,B^{\alpha}$ as a fit variable, the smallest scatter in $T_{\max}$ and $x_{\rm s}$ as functions of $\dot m\,B^{\alpha}$ is, in fact, obtained for $\alpha = -2.6\pm0.2$.

Figure 4: Normalized velocity $w = 4\,v/$ $\upsilon _{\rm o}$ for $M_{\rm wd}$ = 0.6 $M_{\odot }$ and the same values of field strength and mass flow density $\dot m$ as in Fig. 3.

Our model calculations cover magnetic field strengths B = 10-100MG and mass flow densities $\dot m = 10^{-2}{-}10^2$ gcm^-2s^-1. In what follows, we present first the temperature and density profiles along the flow lines. From these, we obtain $T_{\max}$, $x_{\rm s}$, and $h_{\rm sh}$ as the characteristic parameters of the post-shock flow which are presented in an appropriate way as functions of ${\dot m}B^{-2.6}$.

2.8 Electron temperature profiles for an infinite layer

Figure 2 shows the temperature profiles $T_{\rm i}$(x) and $T_{\rm e}$(x) for $M_{\rm wd}$= 0.6 $M_{\odot }$ and several $\dot m$-Bcombinations on a logarithmic depth scale which emphasizes the initial rise of the profiles. These profiles display a substantial spread in $x_{\rm s}$ and in $T_{\max}$, reflecting the influence of cyclotron cooling. At 10^-2gcm^-2s^-1, 100MG, cyclotron cooling has reduced $T_{\max}$ to 6% and $x_{\rm s}$ to 0.3% of the respective values for the pure bremsstrahlung solution. We have confidence in our numerical results because they accurately reproduce the analytic bremsstrahlung solution (see above).

 

Table 1: (a) Top: temperature profiles $T_{\rm e}$/$T_{\max}$(x/$x_{\rm s}$) for a white dwarf mass of 0.6 $M_{\odot }$, field strengths of 10, 30, and 100MG, and mass flow densities between 100 and 0.01gcm^-2s^-1. First line: analytical solution of Eqs. (12) and (15). Subsequent lines: profiles for parameter combinations $B,\,\dot m$ in MG and gcm^-2s^-1, with B₇ in units of 10⁷G. All profiles start at (x/$x_{\rm s}$,$T_{\rm e}$/$T_{\max}$) = (0,0), end at (1,0), and are normalized to a peak value of unity. (b) Bottom: same for the normalized velocity $w = 4\,v/$ $\upsilon _{\rm o}$. These profiles start at (x/$x_{\rm s}$,w) = (0,1) and end at (1,0).

$\dot m$	B	${\dot m}\,B_7^{-2.6}$	T/$T_{\max}$ vs.  x/$x_{\rm s}$
			x/$x_{\rm s}$ = 10-4	10-3	0.01	0.02	0.05	0.10	0.20	0.30	0.40	0.50	0.60	0.70	0.80	0.90	0.95	0.98	0.99
		$\infty$	1.000	0.999	0.995	0.989	0.973	0.945	0.886	0.821	0.751	0.674	0.589	0.494	0.382	0.245	0.156	0.085	0.054
100	10	100	0.230	0.515	0.916	0.984	0.995	0.967	0.905	0.840	0.770	0.691	0.603	0.505	0.392	0.250	0.158	0.085	0.053
10	10	10	0.230	0.515	0.915	0.984	0.996	0.968	0.906	0.842	0.770	0.692	0.604	0.506	0.393	0.250	0.159	0.082	0.048
1	10	1	0.219	0.499	0.898	0.977	0.998	0.973	0.914	0.850	0.782	0.710	0.629	0.536	0.426	0.281	0.176	0.085	0.047
0.1	10	0.1	0.191	0.434	0.822	0.927	0.999	0.980	0.916	0.849	0.779	0.706	0.622	0.522	0.400	0.239	0.127	0.047	0.022
0.01	10	0.01	0.175	0.395	0.743	0.861	0.981	0.991	0.860	0.743	0.650	0.565	0.476	0.384	0.285	0.165	0.092	0.039	0.021
100	30	5.75	0.228	0.512	0.914	0.984	0.994	0.966	0.903	0.836	0.766	0.686	0.598	0.499	0.388	0.247	0.153	0.080	0.046
10	30	0.58	0.224	0.504	0.909	0.981	0.994	0.958	0.888	0.813	0.738	0.657	0.569	0.473	0.361	0.225	0.134	0.065	0.035
1	30	0.058	0.209	0.468	0.865	0.958	0.997	0.937	0.815	0.717	0.629	0.546	0.460	0.372	0.272	0.162	0.091	0.042	0.025
0.1	30	0.0058	0.185	0.408	0.738	0.849	0.971	0.997	0.833	0.657	0.548	0.468	0.401	0.336	0.274	0.197	0.145	0.085	0.045
0.01	30	0.00058	0.178	0.390	0.710	0.802	0.922	0.994	0.878	0.652	0.484	0.362	0.292	0.245	0.196	0.119	0.074	0.038	0.022
100	100	0.25	0.222	0.497	0.898	0.979	0.992	0.942	0.849	0.766	0.685	0.603	0.517	0.424	0.317	0.188	0.104	0.044	0.020
10	100	0.025	0.206	0.468	0.839	0.940	0.999	0.927	0.756	0.639	0.541	0.450	0.367	0.289	0.201	0.103	0.043	0.011	0.006
1	100	0.0025	0.199	0.440	0.795	0.903	0.997	0.921	0.564	0.458	0.375	0.302	0.237	0.193	0.129	0.058	0.027	0.013	0.007
0.1	100	0.00025	0.180	0.354	0.626	0.727	0.880	0.976	0.966	0.600	0.131	0.097	0.084	0.077	0.063	0.047	0.038	0.030	0.022

$\dot m$	B	${\dot m}\,B_7^{-2.6}$	w = 4v/ $\upsilon _{\rm o}$ vs.  x/$x_{\rm s}$
			x/$x_{\rm s}$ = 10-4	10-3	0.01	0.02	0.05	0.10	0.20	0.30	0.40	0.50	0.60	0.70	0.80	0.90	0.95	0.98	0.99
		$\infty$	1.000	0.999	0.992	0.984	0.960	0.921	0.841	0.761	0.678	0.594	0.506	0.413	0.311	0.193	0.121	0.065	0.041
100	10	100	1.000	0.999	0.992	0.985	0.961	0.920	0.840	0.760	0.676	0.593	0.504	0.411	0.310	0.191	0.119	0.064	0.039
10	10	10	1.000	0.999	0.991	0.983	0.958	0.915	0.834	0.755	0.670	0.586	0.496	0.405	0.304	0.187	0.115	0.060	0.035
1	10	1	1.000	0.998	0.983	0.969	0.932	0.882	0.788	0.700	0.618	0.538	0.459	0.376	0.289	0.182	0.112	0.053	0.029
0.1	10	0.1	1.000	0.993	0.934	0.888	0.794	0.696	0.571	0.480	0.404	0.336	0.279	0.218	0.158	0.089	0.046	0.017	0.008
0.01	10	0.01	1.000	0.989	0.892	0.811	0.635	0.440	0.266	0.193	0.150	0.116	0.090	0.066	0.045	0.024	0.013	0.005	0.003
100	30	5.75	1.000	0.999	0.990	0.981	0.953	0.913	0.831	0.751	0.667	0.584	0.497	0.402	0.303	0.186	0.115	0.059	0.033
10	30	0.58	1.000	0.996	0.972	0.948	0.902	0.947	0.758	0.676	0.597	0.517	0.437	0.352	0.264	0.159	0.094	0.044	0.022
1	30	0.058	1.000	0.990	0.914	0.852	0.726	0.614	0.499	0.425	0.363	0.307	0.254	0.199	0.144	0.084	0.047	0.019	0.009
0.1	30	0.0058	1.000	0.990	0.902	0.829	0.664	0.480	0.286	0.206	0.167	0.140	0.117	0.097	0.078	0.055	0.040	0.021	0.011
0.01	30	0.00058	0.999	0.982	0.895	0.824	0.653	0.435	0.148	0.087	0.062	0.045	0.036	0.029	0.023	0.013	0.008	0.004	0.003
100	100	0.25	1.000	0.995	0.948	0.909	0.835	0.760	0.660	0.579	0.506	0.437	0.367	0.294	0.215	0.124	0.068	0.029	0.013
10	100	0.025	1.000	0.989	0.886	0.807	0.643	0.494	0.367	0.302	0.251	0.206	0.167	0.128	0.089	0.045	0.019	0.005	0.002
1	100	0.0025	1.000	0.982	0.860	0.758	0.535	0.282	0.124	0.099	0.081	0.065	0.051	0.040	0.028	0.013	0.006	0.002	0.001
0.1	100	0.00025	0.999	0.985	0.885	0.827	0.707	0.552	0.293	0.069	0.012	0.009	0.008	0.007	0.006	0.004	0.003	0.003	0.002

Figure 3 displays the normalized profiles of the electron temperature, $T_{\rm e}$/$T_{\max}$ vs. x/$x_{\rm s}$, for different $\dot m, B$combinations, covering the range from a bremsstrahlung-dominated flow with 100gcm^-2s^-1, 30MG (fat solid curve) to 10^-2gcm^-2s^-1, 100MG near the non-hydrodynamic limit (dotted curce). They represent an approximate sequence in ${\dot m}B^{-2.6}$, but not surprisingly, the shapes differ somewhat for different $\dot m$ and B combinations with the same value of ${\dot m}B^{-2.6}$ (not shown in Fig. 3).

Equilibration between electron and ion temperatures is reached at column densities of $\sim$10 $^{-3} \ldots 10^{-1}$gcm^-2s^-1 depending on $\dot m$ and B (Fig. 2). At 100gcm^-2s^-1, 10MG, electrons and ions equilibrate as early as $\sim 0.03$$x_{\rm s}$, while at 10^-2gcm^-2s^-1, 100MG, equilibration length and $x_{\rm s}$ are of the same order, indicating the approach to the non-hydrodynamic regime. A peculiar feature of the latter profile is the extended low-temperature tail which was not adequately resolved by WB96. This tail appears when equilibration occurs near the temperature at which cyclotron cooling becomes ineffective and the density is sufficiently high for bremsstrahlung to take over. It is hydrodynamic in origin. Apart from the tail, the temperature profile at 10^-2gcm^-2s^-1, 100MG is very close to that obtained by the non-hydrodynamic approach of WB92, WB93. The low-temperature tail is responsible for a low-temperature thermal emission component with keV.

The initial rise of the individual temperature profiles is similar and is very rapid following approximately $T_{\rm e}$ $\propto x^{0.35}$(Fig.2). One half of $T_{\max}$ is reached at 0.001$x_{\rm s}$ in the bremsstrahlung-dominated case and at 0.006$x_{\rm s}$ near the non-hydrodynamic limit. Further downstream the profiles differ substantially. In the bremsstrahlung-dominated case, the peak electron temperature is reached quickly, while in the cyclotron-dominated flow it occurs at the same x at which half of the accretion energy has been radiated away. The reason is that a temperature gradient is needed to drive about one half of the radiative flux across the shock front, while the other half enters the white dwarf atmosphere. In the plane-parallel geometry, the optically thick radiative transfer requires the electron temperature at the shock front to stay below the peak electron temperature : $T_{\rm e,s} < T_{\rm max} \le T_{\rm brems,max} \sim 0.5\,T_{\rm i,s}$. This is why we opted to start the integration with the initial values $T_{\rm e,s}$ = 0 and $T_{\rm i,s}$ as given by Eq.(7). Because of the rapid initial rise in T(x), our results would have been practically the same had we set $T_{\rm e,s}$= 0.5$T_{\max}$.

To facilitate the modeling of specific geometries, we provide the normalized temperature and density profiles for a sequence of $\dot m, B$ combinations in Table 1. We also provide best fits to $T_{\max}$/ $T_{\rm i,s}$ and $x_{\rm s}$ as functions of ${\dot m}B^{-2.6}$.

Figure 5: Maximum electron temperature $T_{\max}$ as a function of ${\dot m}\,B_7^{-2.6}$ for $M_{\rm wd}$ = 0.6 $M_{\odot }$ and B = 10 - 100MG ( B7 = B/107G). The solid curve is the fit from Eq. (19), the solid straight line represents the bombardment solution from Eq. (18). The dotted curve and straight line represent the corresponding fit to the data for 1 $M_{\odot }$ (data not shown).

Figure 6: Column density $x_{\rm s}$ of the post-shock cooling region as a function of ${\dot m}\,B_7^{-2.6}$ for the same parameters as in Fig. 5. The solid curve is the best fit from Eq. (20). The dotted curve indicates the corresponding fit to the data for 1$M_{\odot }$. The straight lines in the lower left denote the bombardment solutions from Eq. (17) for 0.6 $M_{\odot }$ and 1.0 $M_{\odot }$.

Figure 7: Same as Fig. 6 but for geometrical shock height $h_{\rm sh}$ contained in the quantity $h_{\rm sh}$B72.6. The fits refer to 0.6 $M_{\odot }$ (solid curve) and 1 $M_{\odot }$ (dashed curve). The straight lines represent the corresponding bremsstrahlung solutions of Eq. (14).

Figure 8: Overall spectral energy distributions for an emission region on an 0.6 $M_{\odot }$ white dwarf at d = 10pc with B = 30MG, D = 108cm, and $\dot m = 100,10,1$, and 10-1gcm-2s-1 (from top). The dashed sections indicate the emission directed away from the white dwarf at $\vartheta = 5^{\circ }$without the reflection albedo, the solid curves include the latter. The dotted curve for $\dot m = 1$gcm-2s-1, B = 100MG indicates the increased cyclotron radiation and the reduced bremsstrahlung flux and temperature for this field strength.

2.9 Velocity profiles for an infinite layer

For calculations of the bremsstrahlung emission, we need the profiles of the mass density which varies as $\rho \propto \upsilon ^{-1}$. Figure 4 shows the normalized velocity profiles for $M_{\rm wd}$= 0.6 $M_{\odot }$ and the same $\dot m, B$ combinations as in Fig. 3. In the limit of pure bremsstrahlung cooling, the velocity profile is indistinguishable from that given by the inversion of Eq. (12). Increased cyclotron cooling causes a similar depression at intermediate x as seen in the temperature profiles. Table 1 (bottom) provides the velocity profiles in numerical form for the same parameters as above.

2.10 Maximum electron temperature $\mathsfsl T_{\mathsf {max}}$

In what follows, each model is represented by one "data point''. Figure 5 shows $T_{\max}$/ $T_{\rm i,s}$ vs. ${\dot m}\,B_7^{-2.6}$ for $M_{\rm wd}$=0.6 $M_{\odot }$ and B = 10-100MG. The dependence of $T_{\max}$ on ${\dot m}\,B_7^{-2.6}$ is equally well documented for $M_{\rm wd}$=0.8 and 1.0 $M_{\odot }$, but for clarity we do not show these data. The 0.6 $M_{\odot }$ results can be fitted by

$$\frac{1}{T_{\rm max}} = \left[\left(\frac{1}{a_0\,T_{\rm max,... ...c{1}{a_1\,T_{\rm max,brems}}\right)^\alpha ~\right]^{1/\alpha}$$ (19)

with $T_{\rm max,bomb}$ from Eq. (18) and $T_{\rm max,brems}$ from Eq. (16). The exponent $\alpha$ measures the smoothness of the transition between cyclotron and bremsstrahlung solutions. The fits for 0.6 and 1 $M_{\odot }$ are included in Fig. 5 as the solid and dotted curves, respectively. The fit parameters a₀, a₁, and $\alpha$ are listed in Table 2 for all three white dwarf masses. The fact that a₁ falls slightly short of 1.0 indicates that the limiting value $T_{\rm max} = (\mu/\mu_{\rm i})T_{\rm i,s}$ for large $\dot m$ is not yet reached at 100gcm^-2s^-1. Even at this high $\dot m$, radiative energy losses remove some energy prior to equipartition. The maximum temperatures at low $\dot m$ remain about 10% below the temperatures predicted by the bombardment solution of WB92,93 (straight lines, $a_0 \simeq 0.9$). A difference as small as this is actually remarkable considering the substantially different theoretical and numerical approaches (radiation hydrodynamics vs. radiative transfer in a static atmosphere).

 

 

Table 2: Fit parameters of Eqs. (19)-(21) for three values of the white dwarf mass $M_{\rm wd}$.

M	a0	a1	$\alpha$	b0	$\beta$	c0	$\gamma$
($M_{\odot }$)				(s)		(108cm)
0.6	0.91	0.968	1.67	6.5	0.70	0.95	1.0
0.8	0.86	0.954	1.54	7.5	0.54	1.30	0.7
1.0	0.90	0.934	1.25	8.0	0.45	1.75	0.5

2.11 Column density $\mathsfsl x_{\mathsf s}$

The transition of $x_{\rm s}$ between the bombardment and the bremsstrahlung solutions (Eqs. (17) and (13)) is more complicated than that of $T_{\max}$. Figure 6 shows $x_{\rm s}$ as a function of ${\dot m}\,B_7^{-2.6}$ for $M_{\rm wd}$ = 0.6 $M_{\odot }$ and B = 10 - 100MG. Again, the data points for 0.8 and 1.0 $M_{\odot }$ are not shown for clarity. We fit $x_{\rm s}$ by

$$\frac{1}{x_{\rm s}} = \left[\left(\frac{\mu}{b_0\,\mu_{\rm e}... ... \left(\frac{1}{x_{\rm s,brems}}\right)^\beta\right]^{1/\beta}$$ (20)

with $x_{\rm s,brems}$ from Eq. (13). The fit parameters b₀ and $\beta$ are given in Table 2 for the three values of $M_{\rm wd}$. Again, the fits are shown for 0.6 and 1.0 $M_{\odot }$ (solid and dotted curve), with the corresponding bombardment solutions added as straight lines. Note that, contrary to what we found for $T_{\max}$, the first term in Eq. (20) does not represent the bombardment solution, but rather the cyclotron-dominated shock heated plasma. It connects to the bombardment solution as the non-hydrodynamic regime is approached and bridges a gap of two orders of magnitude in $x_{\rm s}$ between the bombardment and bremsstrahlung solutions. Clearly, the quantitative determination of $x_{\rm s}(\dot m, B)$ requires radiation-hydrodynamical calculations. The $\dot m$-dependence of Eq. (20) reproduces that of Eq. (10), the molecular weight dependence is added here and taken from Eq. (9).

Figure 9: Cyclotron section of the spectral energy distributions for the same set of parameters as in Fig. 8, except $\vartheta = 80^{\circ }$, and for mass flow densities $\dot m = 1, 10^{-1}$, and 10-2gcm-2s-1 (from top).

2.12 Geometrical shock height $\mathsfsl h_{\mathsf {sh}}$

Figure 7 shows the quantity $h_{\rm sh}$B₇^2.6 for $M_{\rm wd}$ = 0.6 $M_{\odot }$ and for field strengths between 10 and 100MG. We fit the data by

$$\frac{1}{h_{\rm sh}B_7^{2.6}} = \left[\left(\frac{\mu}{c_0\,\... ..._7^{-2.6}}{h_{\rm sh,brems}}\right) ^\gamma~\right]^{1/\gamma}$$ (21)

where $\dot m$ as independent variable enters via $h_{\rm sh,brems}$from Eq. (14) and the fit parameters c₀ and $\gamma$are listed in Table 2. The fits for 0.6 $M_{\odot }$ and 1.0 $M_{\odot }$ are shown (solid and dotted curve). The limiting dependencies for large and small ${\dot m}B^{-2.6}$, respectively, are $h_{\rm sh}$ $\, = h_{\rm sh,brems} \propto \dot m^{-1}$ and $h_{\rm sh}$= $c_0\,B_7^{-2.6}$ = const., as predicted by Eq. (11). The shock height is related to $x_{\rm s}$ by the mean post-shock density $\overline{\rho}$ = $x_{\rm s}$/ $h_{\rm sh}$ which is $\overline{\rho} = 6.97\,\rho_0$ for the bremsstrahlung-dominated shock solution (see Eqs. (13) and (14)). For the cyclotron-dominated shock-heated flow, the mean post-shock density increases to $\overline{\rho} = (b_0/c_0)\,\rho_0\,\upsilon_{\rm o} \simeq 30\,\rho_0$, a result which cannot be obtained from simple theory.