A&A 373, 211-221 (2001)
DOI: 10.1051/0004-6361:20010600
Universitäts-Sternwarte, Geismarlandstr. 11, 37083 Göttingen, Germany
Received 30 November 2000 / Accepted 23 April 2001
Abstract
We have solved the one-dimensional stationary two-fluid
hydrodynamic equations for post-shock flows on accreting magnetic
white dwarfs simultaneous with the fully frequency and angle-dependent
radiative transfer for cyclotron radiation and bremsstrahlung.
Magnetic field strengths B = 10 to 100MG are considered. At given
B, this theory relates the properties of the emission region to a
single physical parameter, the mass flow density (or accretion rate
per unit area) .
We present the normalized temperature
profiles and fit formulae for the peak electron temperature, the
geometrical shock height, and the column density of the post-shock
flow. The results apply to pillbox-shaped emission regions. With a
first-order temperature correction they can also be used for narrower
columns provided they are not too tall.
Key words: stars: cataclysmic variables - stars: white dwarfs -stars: binaries: close - radiative transfer -hydrodynamics
The thermal structure of the accretion columns on accreting magnetic
white dwarfs can be derived analytically for a single-particle fluid
and sufficiently simple assumptions on the radiative cooling (Aizu
1973; Chevalier & Imamura 1982; Wu et al. 1994). For the more
general case of the optically thick frequency and angle-dependent
radiative transfer in a two-fluid plasma, the coupled hydrodynamic and
radiative transfer equations have to be solved numerically (Woelk &
Beuermann 1996, henceforth WB96). In this paper, we present results
which are improved and expanded over those of WB96. We obtain the
temperature and density profiles for plane-parallel post-shock cooling
flows and derive fit formulae for the peak electron temperature ,
the column density
,
and the geometrical shock height
as
functions of the magnetic field strength B and the mass flow density
(accretion rate per unit area)
.
For low
and high
B, we show that the shock solution merges into the non-hydrodynamic
bombardment solution for an atmosphere which is heated by a stream of
fast ions and cools by cyclotron radiation (Woelk & Beuermann 1992,
1993, henceforth WB92, WB93).
Our treatment of radiation-hydrodynamics is onedimensional and
stationary. The one-dimensionality implies that our solutions are
strictly applicable only to pillbox-shaped emision regions with a
width
and a stand-off distance
,
where
is the white dwarf radius. The stationarity implies that our
solutions describe the mean properties of the shocks and that aspects
like rapid fluctuations in the mass flow density and the stability
against shock oscillations are left aside. Shock oscillations have
been treated by a number of authors (Imamura et al. 1996; Saxton & Wu
1999, and references therein) and generally suggest that cyclotron
cooling stabilizes the flow and bremsstrahlung cooling destabilizes
it. Observationally, optical oscillations have been found in a few
polars, while the search for hard X-ray oscillations has so far
yielded only upper limits (Larsson 1992; Wolff et al. 1999;
Imamura et al. 2000, and references therein).
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)
The connecting link between the hydrodynamics (Eqs. (1) to (4)) and the
radiative transfer (Eqs. (5) and (6)) is
:
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
(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
including the low-
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
/
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
.
Settling solutions with
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
cm, where
is the electron temperature
at the shock in K (Imamura et al. 1987) and
is in gcm-2s-1. Near
the one-fluid limit, electron and ion shock temperatures are similar,
,
and the precursor extends to
.
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
= 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
,
the bulk velocity to
/4, and the pressure to
(3/4)
2, with
and
the density and bulk
velocity in the pre-shock flow. With
= 0, the ion shock
temperature is
All numerical calculations are performed for a hydrogen plasma with
and
,
where
is the number of nucleons per electron,
is the
molecular weight of all particles, and
the molecular
weight of the ions weighted with
.
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
,
,
,
and
.
The frequency-integrated volume emissivity for brems- strahlung is
![]() |
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. |
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Figure 1 shows the schematic of an emission region with finite lateral
extent D. The shock is located at
above the white
dwarf surface. The downstream column density is x = 0 at the shock
and and
at the surface of the star, with x and hbeing related by d
dh. The gravity vector
and
the magnetic field vector
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,
and
.
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
of van Teeseling et al. (1994). The
fraction
of the energy is re-emitted in the UV and soft
X-ray regime and is not considered in this paper.
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 ,
low B limit is the
bremsstrahlung-dominated one-fluid solution. In the opposite limit of
low
,
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),
,
and g = 0 yields
which allows to express the
emissivities of Sect. 2.2 as
and
,
with f and g being functions of the flow
velocity
and with additional dependencies on the
's
and B contained in the proportionality factors. Integration of the
energy equation over
yields expressions for the column
density
and the geometrical shock height
which reflect the
parameter dependence of
,
![]() |
Figure 2:
Temperature profiles for the ions (dashed
curves) and electrons (solid curves) as functions of column density
x for
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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For a strong shock in a one-fluid plasma with adiabatic index 5/3, the
normalized post-shock velocity
varies between
1 and 0. The column density x measured from the shock is related to
by (Aizu 1973; Chevalier & Imamura 1982)
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
appears in
:
With increasing ,
a shock develops which is initially
cyclotron-dominated and ultimately bremsstrahlung-dominated. Since
reaches
at some intermediate
,
we
expect a smooth transition in peak temperature between these cases.
The situation is quite different for
,
however. At the
where
equals
,
and
differ by more
than two orders of magnitude. The run of
(
)
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
2.
![]() |
Figure 3:
Normalized electron temperature distributions for
![]() ![]() ![]() |
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In the bombardment solution (Eqs. (17), (18)), the dependence of
and
on
is obtained from the equilibrium between the
energy gain by accretion,
,
and the
energy loss by optically thick cyclotron radiation,
,
where
is the high-frequency cutoff of the cyclotron spectrum and
the cyclotron frequency. We determine the
limiting harmonic number m* from the cyclotron calculations of
Chanmugam & Langer (1991; their Fig. 5) as
with
and
.
This approximation is valid near depth parameters
and temperatures
and is more adequate for the
cyclotron-dominated emission regions on polars than the frequently
quoted formula of Wada et al. (1980). Replacing
with
in
and equating the
accretion and radiative energy fluxes yields the result that a
power of
is proportional to
.
The same holds for
.
We find that the cyclotron-dominated shocks at low
behave
similarly to bombarded atmospheres in that their thermal properties,
too, depend on
.
The individual temperature profiles
for different
with the same
coincide only
in an approximate way, but the dependency on
holds quite well
for the two characteristic values of each profile,
and
.
If we
leave the exponent
in
as a fit variable,
the smallest scatter in
and
as functions of
is, in fact, obtained for
.
![]() |
Figure 4:
Normalized velocity ![]() ![]() ![]() ![]() ![]() |
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Figure 2 shows the temperature profiles (x) and
(x) for
= 0.6
and several
-Bcombinations on a logarithmic depth scale which emphasizes the initial
rise of the profiles. These profiles display a substantial spread in
and in
,
reflecting the influence of cyclotron cooling. At
10-2gcm-2s-1, 100MG, cyclotron cooling has reduced
to 6%
and
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).
![]() |
B |
![]() |
T/![]() ![]() |
||||||||||||||||
x/![]() |
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 | |||
![]() |
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 |
![]() | B |
![]() | w = 4v/
![]() ![]() |
||||||||||||||||
x/![]() | 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 | |||
  | ![]() | 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, /
vs. x/
,
for different
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
,
but not surprisingly, the shapes
differ somewhat for different
and B combinations
with the same value of
(not shown in Fig. 3).
Equilibration between electron and ion temperatures is reached at
column densities of 10
gcm-2s-1 depending on
and B (Fig. 2). At 100gcm-2s-1, 10MG, electrons and ions
equilibrate as early as
,
while at 10-2gcm-2s-1,
100MG, equilibration length and
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
(Fig.2). One half of
is reached at 0.001
in
the bremsstrahlung-dominated case and at 0.006
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 :
.
This is why we opted to start the
integration with the initial values
= 0 and
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
=
0.5
.
To facilitate the modeling of specific geometries, we provide the
normalized temperature and density profiles for a sequence of
combinations in Table 1. We also provide best fits to
/
and
as functions of
.
![]() |
Figure 5:
Maximum electron temperature ![]() ![]() ![]() ![]() ![]() |
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![]() |
Figure 6:
Column density ![]() ![]() ![]() ![]() ![]() |
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![]() |
Figure 7:
Same as Fig. 6 but for geometrical
shock height
![]() ![]() ![]() ![]() |
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![]() |
Figure 8:
Overall spectral energy distributions for an
emission region on an 0.6 ![]() ![]() ![]() ![]() |
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For calculations of the bremsstrahlung emission, we need the profiles
of the mass density which varies as
.
Figure 4 shows the normalized velocity profiles for
=
0.6
and the same
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.
In what follows, each model is represented by one "data point''.
Figure 5 shows /
vs.
for
=0.6
and
B = 10-100MG. The dependence of
on
is equally well
documented for
=0.8 and 1.0
,
but for clarity we do not
show these data. The 0.6
results can be fitted by
M | a0 | a1 | ![]() |
b0 | ![]() |
c0 | ![]() |
(![]() |
(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 |
The transition of
between the bombardment and the bremsstrahlung
solutions (Eqs. (17) and (13)) is more complicated
than that of
.
Figure 6 shows
as a function of
for
= 0.6
and
B = 10 - 100MG. Again, the data points for
0.8 and 1.0
are not shown for clarity. We fit
by
![]() |
Figure 9:
Cyclotron section of the spectral energy
distributions for the same set of parameters as in Fig. 8, except
![]() ![]() |
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Figure 7 shows the quantity
B72.6 for
=
0.6
and for field strengths between 10 and 100MG. We fit
the data by
In our radiation-hydrodynamical calculations, energy conservation is
enforced and the radiative luminosity of the infinite layer per unit
area equals the accretion energy
.
Any real emision region,
however, has a finite lateral width and looses energy not only from
its top and bottom surfaces but also from its sides (Fig. 1).
Two-dimensional radiation hydrodynamics would then be needed to
calculate the temperature structure. In this section we discuss to
what extent our one-dimensional results are still applicable to
regions of finite extent.
We consider an emission region as depicted in Fig. 1, with finite
width D, cross section D2, and field strength B. In a first
step, we adopt the temperature and velocity profiles along the flow
lines, T(x) and
,
calculated for infinite D also for
the case of finite D.
In the Rybicki code the radiation transport equation was solved with a
mean cyclotron absorption coefficient and electron scattering was
included. For the emission region of finite extent, we calculate the
outgoing flux at angle
by ray tracing using the
temperature profiles along slanted paths as shown schematically in
Fig. 1. For rays starting or ending on the side surfaces, the
temperature and density profiles were truncated appropriately. We
account separately for the cyclotron emissivities in the ordinary ray
(index o) and the extraordinary ray (index e), and add 50% of the
free-free emissivity with Gaunt factor to both. We neglect electron
scattering in the emission region, but include the atmospheric albedo
(van Teeseling et al. 1994). Each ray yields a contribution
to the integrated intensity
(in ergs-1Hz-1sr-1) in that direction and the
summation is extended over n rays,
Figure 8 shows the spectral flux at
against the field direction emitted by an emission region with B =
30MG and an area of 1016cm2 (D = 108cm) on an
0.6
white dwarf at a distance of 10 pc. Cyclotron emission
dominates for low
and bremsstrahlung for high
.
Free-free absorption becomes important near 1015Hz at the
highest
,
but in reality this spectral region is dominated by
the quasi-blackbody component produced by reprocessing of the incident
flux in the white dwarf atmosphere. The results of WB96 on the ratio
of the cyclotron vs. bremsstrahlung luminosities as a function of
and B remain basically valid, but will be modified if the
shock is buried in the atmosphere and X-ray absorption is accounted
for.
Figure 9 illustrates the optical depth dependence of the
cyclotron spectra at
.
Cyclotron emission
lines at low
change into absorption features at high
.
Since in real emission regions the fractional area of the
high-
section is small (Rousseau et al. 1996) observed spectra
show emission lines.
An isolated emission region of lateral width D, shock height
,
and the temperature profile T(x) of the infinite layer appropriate
for the mass flow density
will have
for optically thin and
for optically thick
emission. The overestimate in the latter case results from radiation
emerging from the sides of the region without a compensating
influx. For the optical depths considered here, bremsstrahlung is
practically free of such overestimate, cyclotron radiation is not.
Let us assume for the infinite layer that
feeds two components of
,
namely
and
.
For finite D, we then have
![]() |
Figure 10: Quantity A from Eq. (24) measuring
the excess luminosity
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Open with DEXTER |
In order to assess the size of the possible error for AMHer stars,
we estimate
/D for a typical accretion rate of
gs-1 as a function of
.
The linear width of
the emission region is
cm with
in gcm-2s-1. For a
bremsstrahlung-dominated flow, Eq. (14) yields
(for
= 0.6
)
which is less than unity since in this case
gcm-2s-1. For
a cyclotron-dominated flow, Eq. (21) correspondingly yields
which is
again less than unity since now
gcm-2s-1 and typically
B7 > 1. Hence,
is seriously overestimated only for
isolated narrow subcolumns which are not radiatively shielded by
neighboring fluxtubes.
Let the application of the unmodified one-dimensional temperature
profile T(x)yield a specific luminosity
with r > 1. We can then take then either: (i)
identify the parameters of this emission region with those appropriate
for the increased mass flow rate
;
or (ii) recalculate the emission for a reduced mass flow rate
and identify temperature and emission of that region
as appropriate for the initial
.
In case (ii),
and
are reduced to
(
)
and
(
). This approach
demonstrates that the rising sections of the relations displayed in
Fig. 5 (Eq. (19)) and Fig. 6 (Eq. (20))
are further depressed for narrow columns, while the horizontal parts,
where optically thin bremsstrahlung dominates, are not affected. Both
approaches secure energy conservation but cannot replace a proper
treatment of the problem. They are not recommended for isolated tall
columns.
We have solved the equations of one-dimensional, two-fluid stationary
radiation hydrodynamics for the shock-heated plasma in the emission
regions on accreting magnetic white dwarfs for a wide range of mass
flow densities
and field strengths
B. For given B and
,
the peak electron temperature
and
the column density
of the emission region are physically related
to
as the independent variable of the theory. They are no
longer independent variables as in the frequently employed
"constant-
models''. It is possible, therefore, to
interpret the observed spectral energy distributions of accreting
magnetic white dwarfs in terms of the distribution of mass flow
densities present in their accretion spots.
We now discuss to what extent the application of these results is
limited by the simplifications made in our calculations. One major
simplification is the assumption of stationarity which implies that we
neglect the possible occurrence of shock oscillations (Imamura et al. 1996; Saxton & Wu 1999, and references therein) and that we can
not treat rapid time variability of .
Since our approach can
accommodate a range of
to occur in neighboring columns, the
emitted spectrum will still approximate the true time-averaged
spectrum if
varies only on time scales exceeding the
post-shock cooling time
s. In the presence of shock
oscillations which have periods of order
,
our
results yield a mean temperature and column density which need not
agree with the true time-averaged value if the oscillation is
nonlinear (Imamura & Wolff 1990).
The assumption of a one-dimensional flow implies that we neglect the
convergence of the polar field lines of the white dwarf. In the
spirit of this approximation, we have included the acceleration of the
post-shock flow by a constant gravity G
/
2 and
neglected the r-dependence of g.
The assumption of an infinite layer implies that there is no
temperature gradient perpendicular to the flow. This is no restriction
for bremsstrahlung-dominated flows, but in such gradient is always
established in columns of finite width D by optically thick
radiation components and lowers the mean electron temperature averaged
across the column at any position x. We have suggested a simple
first-order correction for the implied overestimate in T(x) which
ensures conservation of energy and provides some remedy for narrow
columns with
.
For very narrow columns with
or absolutely tall columns with
,
the main radiative energy flow may be
sideways and the approach of Wu et al. (1994) becomes more
appropriate. In summary, our results are valid whenever
and
.
On the positive side, we consider our largely correct treatment of the
two-fluid nature of the post-shock flow. One-fluid treatments
(e.g. Chevalier & Imamura 1982; Wu et al. 1994) can account for
cooling by cyclotron radiation in addition to bremsstrahlung, but are
limited, by definition, to mass flow densities sufficiently high to
ensure quick equilibration of electron and ion temperatures. They can
not describe the substantial reduction of the peak electron
temperature below the one-fluid value which we show to be present at
low mass-flow density
and/or high magnetic field strength
B. As a result, our description catches the essential properties of
such columns: (i) dominant cyclotron cooling causes the peak
electron temperature to stay far below the peak temperature of the
one-fluid approach; (ii) cyclotron cooling causes a drastic reduction
in the column density and the geometrical shock height of the
post-shock flow compared with pure bremsstrahlung cooling; and (iii)
peak temperature and column density vary smoothly between the two
limiting cases, the bremsstrahlung-dominated high-
regime
(Aizu 1973) and the cyclotron-dominated low-
bombardment
solution (WB92, WB93). The latter denotes the transition to the
non-hydrodynamic regime and, gratifyingly, our calculations recover
the bombardment solution at the lowest mass flow densities
accessible. Compared with WB96, we obtained numerically more accurate
results and have cast these into simple-to-use fit formulae which
facilitate the modeling of emission regions within the geometrical
limitations noted above. No other two-fluid calculations with the full
optically thick radiative transfer are available.
The remaining, mainly geometrical limitations of our approach are inherently connected to the one-dimensional radiative transfer. Extension of the calculation to two dimensions encounters two problems: (i) a substantial increase in complexity and (ii) the introduction of an additional free parameter in form of the lateral width of the emission region. Therefore, we consider our one-dimensional approach with the correction explained in Sect.3 as a reasonable compromise, with the noted exception of tall columns.
The present results can be used to quantitatively model the emission
regions on accreting magnetic white dwarfs. The discussion of the
overall spectral energy distribution of such objects requires to
account for shocks being buried in the photosphere of the white
dwarf. Such model allows to obtain the distribution in the
accretion spot from observational data and is presented in Paper II of
this series. The emission properties of AM Herculis binaries depend
not only on
but vary also systematically with field strength:
this dependence is described in Paper III.
Acknowledgements
This work is based on a code originally devised by U. Woelk. We thank B. T. Gänsicke, F. V. Hessman and K. Reinsch for numerous discussions and the referee J. Imamura for helpful comments which improved the presentation of the results. This work was supported in part by BMBF/DLR grant 50OR99036.