Issue 
A&A
Volume 571, November 2014



Article Number  A55  
Number of page(s)  14  
Section  Stellar atmospheres  
DOI  https://doi.org/10.1051/00046361/201423724  
Published online  06 November 2014 
Modeling the EUV spectra of optically thick boundary layers of dwarf novae in outburst
^{1}
Institut für Astronomie und Astrophysik, Kepler Center for Astro and
Particle Physics, Universität Tübingen,
Sand 1,
72076
Tübingen,
Germany
email:
suleimanov@astro.unituebingen.de
^{2}
Kazan (Volga region) Federal University,
Kremlevskaja str. 18,
420008
Kazan,
Russia
Received: 27 February 2014
Accepted: 14 August 2014
Context. Disk accretion onto weakly magnetized white dwarfs (WDs) in cataclysmic variables (CVs) leads to the formation of a boundary layer (BL) between the accretion disk and the WD, where the accreted matter loses its excess kinetic energy and angular momentum. It is assumed that angular momentum is effectively transported in the BL, but the transport mechanism is still unknown.
Aims. Here we compute detailed model spectra of recently published optically thick onedimensional radial BL models and qualitatively compare them with observed soft Xray/extreme ultraviolet (EUV) spectra of dwarf novae in outburst.
Methods. Every considered BL model with given effective temperature and surface density radial distribution is divided into a number of rings, and for each ring, a structure model along the vertical direction is computed using the stellaratmosphere method. The ring spectra are then combined into a BL spectrum taking Doppler broadening and limb darkening into account.
Results. Two sets of model BL spectra are computed, the first of them consists of BL models with fixed WD mass (1 M_{⊙}) and various relative WD angular velocities (0.2, 0.4, 0.6 and 0.8 breakup velocities), while the other deals with a fixed relative angular velocity (0.8 breakup velocity) and various WD masses (0.8, 1, and 1.2 M_{⊙}). The model spectra show broad absorption features because of blending of numerous absorption lines, and emissionlike features at spectral regions with only a few strong absorption lines. The model spectra are very similar to observed soft Xray/EUV spectra of SS Cyg and U Gem in outburst. The observed SS Cyg spectrum could be fitted by BL model spectra with WD masses 0.8−1 M_{⊙} and relative angular velocities 0.6−0.8 break up velocities. These BL models also reproduce the observed ratio of BL luminosity and disk luminosity. The difference between the observed and the BL model spectra is similar to a hot optically thin plasma spectrum and could be associated with the spectrum of outflowing plasma with a mass loss rate compatible with the BL mass accretion rate.
Conclusions. The suggested method of computing BL spectra seems very promising and can be applied to other BL models for comparison with EUV spectra of dwarf novae in outburst.
Key words: accretion, accretion disks / stars: dwarf novae / radiative transfer / methods: numerical / Xrays: binaries
© ESO, 2014
1. Introduction
The importance of energy release between an accretion disk and a central object with a surface was realized almost immediately (LyndenBell & Pringle 1974) after the introduction of modern accretion disk theory (Shakura & Sunyaev 1973). Later a similar onedimensional (1D) boundarylayer (BL) theory was developed (Pringle 1977; Pringle & Savonije 1979; Tylenda 1981; Regev 1983; Bertout & Regev 1992; Regev & Bertout 1995; Godon et al. 1995; Popham & Narayan 1995). Here we mainly consider BLs around white dwarfs (cataclysmic variable stars (CVs), see review in Warner 2003) and describe a few key points of BL theory.
Depending on the accretion rate, Ṁ, a BL can be optically thin (Ṁ< 10^{16} g s^{1}) or optically thick (Ṁ> 10^{16} g s^{1}) (Pringle & Savonije 1979). The BL optical thickness also depends on the mass and angular velocity of the white dwarf (WD), as well as on the value of turbulent viscosity in the BL (Popham & Narayan 1995; Collins et al. 2000a). For the description of optically thick BLs, two qualitatively different approaches were suggested, which reduced the problem to a 1D model. In the first approach, the BL is considered as the inner part of the 1D axisymmetric accretion disk (Pringle & Savonije 1979; Regev 1983; Popham & Narayan 1995) with no vertical component of the velocity. The condition imposed on the BL at the inner boundary is that it rotates at the stellar equatorial velocity. Therefore, the accreting matter has to decelerate and release its excess energy in a relatively narrow (a few percent of the inner disk radius) ring. Most of this energy is radiated away, but part of it can be advected into the WD (Popham 1997; Godon 1997), deposited to the outflow (Mauche & Raymond 2000), and can accelerate the outer layers of the WD forming a fast rotating belt (Long et al. 2006). In contrast, in the second approach, the matter keeps its nearly Keplerian velocity at the central object’s equator and spreads over the surface (Inogamov & Sunyaev 1999). The spreading matter loses its kinetic energy gradually due to friction with the more slowly rotating surface of the central star and radiates the released energy mainly in two bright highlatitude belts. Initially the spreading layer model was developed for BLs around neutron stars (Inogamov & Sunyaev 1999), but later it was extended to the WD case, too (Piro & Bildsten 2004b).
Together with the difference in geometry, the description of the viscosity, which provides a strong coupling of rings of matter moving with different velocities and an effective thermal dissipation of the lost energy, is different in both approaches. In the first model the usual α prescription for accretion disks is used, which means that the rϕcomponent of the viscosity stress tensor is parameterized by the total pressure P at a given point as w_{rϕ} = αP (Shakura & Sunyaev 1973) or, almost equivalently, a similar parametrization of kinematic viscosity, ν_{α}, is used. In the second approach the α viscosity is completely ignored and only the friction between the highvelocity spreading matter and the dense and relatively cool stellar envelope is considered. The corresponding component of the specific frictional force, , is again scaled using the matter density at the bottom of the spreading layer, ρ_{b}, and the relative spreading matter velocity, v_{sl} (Inogamov & Sunyaev 1999). The estimated value of α_{b} is relatively low, about 10^{3} (Inogamov & Sunyaev 1999). Detailed consideration of this problem shows that the matter deceleration due to friction with the underlying stellar envelope is not trivial and has to be investigated more carefully (Inogamov & Sunyaev 2010).
It is clear from both viscosity descriptions that we do not know the physics of kinetic energy loss and angular momentum transfer in BLs (see, e.g., Kato & Inagaki 1994; Narayan et al. 1994; Godon 1995a). The magnetorotational instability (MRI; Velikhov 1959; Chandrasekhar 1960; Balbus & Hawley 1991), which is usually considered as a physical realization of the α viscosity, cannot operate in the case of angular velocity decrease taking place in BLs (Godon 1995a). Recently, Belyaev & Rafikov (2012) and Belyaev et al. (2012, 2013) have suggested a new physical model for an angular momentum transport in BLs based on acoustic instabilities. Various hydrodynamical instabilities, beginning with simple shear instabilities (Kippenhahn & Thomas 1978), were also considered before as a way of angular momentum transport (see the review in Belyaev & Rafikov 2012).
Another important feature of BLs is their 3D nature. A full 3D treatment is important to account for a correct turbulent viscosity description. But BL models that use a parametrization of the turbulent viscosity could be considered as axisymmetric 2D models. The first attempts to describe the BL around the WD using a 2D timedependent hydrodynamical approach were performed many years ago (Robertson & Frank 1986; Kley & Hensler 1987; Kley 1989). The importance of the viscosity prescription was also demonstrated (Kley 1991). Recently, this kind of computation was repeated with higher, albeit insufficient temporal and spatial resolution without (Fisker & Balsara 2005) and with rudimentary radiation treatment (Balsara et al. 2009). These investigations have confirmed that matter is spreading over the WD surface for optically thick BLs.
Astrophysical observations give the possibility to constrain the correct description of angular moment transport and energy dissipation in BLs. It is necessary to compare properties of a mature BL model with observed features. There are two ways for this kind of comparison. The first one is to investigate flux variability, which is probably connected with BLs, using a noise power spectrum approach, for instance van der Klis (1989). There are a lot of observational data about rapid flux variability in lowmass Xray binaries (LMXBs), such as quasiperiodic oscillations (QPOs) and power density spectra (van der Klis 2000). The cumulative data about rapid variability of CVs, especially in the optical band, is even larger (see, e.g., Patterson 1981; Warner 1986, 2003; Warner & Pretorius 2008,and references therein). Two types of rapid oscillations were distinguished (Robinson & Nather 1979), namely highdegree coherent oscillations with relatively short periods (~7−70 s) in dwarf nova outbursts (DNOs), and less coherent quasiperiodical oscillations (QPOs) with longer periods (up to tens of minutes). The amplitudes of variability for both DNOs and QPOs are relatively low in the optical band (<0.01 mag). On the other hand, the amplitude of these oscillations in Xrays can be much greater (tens of percent, Cordova et al. 1984; Jones & Watson 1992). It is interesting that the coherent oscillations in SS Cyg are observed in soft Xrays, but not in hard Xrays (Swank 1979; Jones & Watson 1992). This fact supports the hypothesis that the coherent oscillations might be connected with an optically thick BL that could be responsible for the soft Xray radiation of SS Cyg during outburst. There are a few relatively simple models to explain DNOs and QPOs (see, e.g., Popham 1999; Warner & Woudt 2002; Piro & Bildsten 2004a; Godon 1995b; Collins et al. 1998, 2000b), and part of them are connected with the existing BL models.
The second way is a comparison of observed emergent spectra of close binary systems with predicted spectra of accretion disks and BLs. Emergent spectra could be computed for BL models with the simple local αviscosity, because those models consider energy release and predict some bolometric radiation flux distribution over the BL. BL model spectra in blackbody approximation were computed by many authors (see, e.g. Tylenda 1977; Kley 1991; Popham & Narayan 1995). The contribution of BL radiation to the FUV spectra of some CVs was also taken into account by computing models of BL rings using the stellaratmosphere method (Godon & Sion 2011; Godon et al. 2012).
Luminous LMXBs (with L> 0.05−0.1 L_{Edd}) with neutron stars show relatively soft and wide Xray spectra, which can be represented to first approximation by two blackbodies with temperatures kT about 1 keV and 2.5 keV (Mitsuda et al. 1984). It is possible to assume that the first component corresponds to the accretion disk, and the second one corresponds to the BL. Study of Xray variability helped to distinguish between these two components (Gilfanov et al. 2003; Revnivtsev & Gilfanov 2006). These authors extracted the spectra of the strongly variable component of the observed Xray flux in a few LMXBs and found that they are very similar for all the investigated sources and that they can be described by a comptonized plasma spectrum with kT ≈ 2.4 keV and electron scattering optical depth τ_{e} ≈ 5−7. They connected these highvariability components of the Xray flux to the BL emission. The spectra of these components were described by model spectra of spreading layers (Suleimanov & Poutanen 2006). Another reference to the validity of the hypothesis that the BLs around neutron stars are spreading layers was presented by Revnivtsev et al. (2013). The highvariability spectral component of the Xray transient source XTE J1701462 keeps the same spectral shape when the bolometric luminosity varies by a factor of twenty. The maximum temperature of this component (Wien tail color temperature) is about 2.4−2.6 keV and coincides with the maximum color temperature of the type I Xray bursts of the same source. It is believed that the latest color temperature of a burst corresponds to the Eddington luminosity (see, e.g., review Lewin et al. 1993). Therefore, most probably, the BLs in LMXBs consist of a part of the neutron star surface that radiates at the Eddington limit with the emitting area being proportional to the luminosity, as it was predicted by the spreading layer model (Inogamov & Sunyaev 1999).
Xray and extreme ultraviolet (EUV) radiation of CVs, which can be associated with BL emission, is much more diverse and a univocal interpretation is difficult (see, e.g., Patterson & Raymond 1985a,b). Blackbody approximation of the optically thick BL model spectra predicts a luminous soft Xray/EUV component with temperature 200−500 kK (Pringle & Savonije 1979; Popham & Narayan 1995; Collins et al. 2000a; Piro & Bildsten 2004b; Hertfelder et al. 2013) in the spectra of all high massaccretion rate CVs without significant magnetic field (dwarf novae in outburst and novalike stars). In fact, only a few of them have this kind of component in their spectra, e.g. SS Cyg and U Gem in outburst (Cordova et al. 1984). The Xray spectra of other nonmagnetic CVs are rather hard (van Teeseling & Verbunt 1994; van Teeseling et al. 1996) and can be described by the model of a cooling flow with temperatures kT from a few keV up to tens of keV (Done et al. 1995; Mukai et al. 2003; Baskill et al. 2005). This kind of radiation is natural for optically thin BLs (Pringle & Savonije 1979), or even for the inner hot accretion flow (Revnivtsev et al. 2012) to be expected for dwarf novae in quiescence, but requires some nontrivial explanation for high massaccreting CVs, like V603 Aql (see, e.g. Patterson & Raymond 1985b). Formally, Popham & Narayan (1995) have found that the transition to the optically thin regime in the model BLs could be reached at sufficiently high mass accretion rates. But their results are questionable because they underestimated the Rosseland “true” opacity by at least two orders of magnitude, using Kramers opacity k_{kr} = k_{0}ρT^{3.5} cm^{2} g^{1} with k_{0} = 6.6 × 10^{22}. The more realistic coefficient is much larger, k_{0} = 5 × 10^{24} (Frank et al. 2002).
At present time, soft Xray/EUV components were found in four dwarf novae in (super) outbursts: SS Cyg (Cordova et al. 1980a, 1984; Mauche et al. 1995; Mauche 2004), U Gem (Cordova et al. 1980b, 1984; Long et al. 1996), VW Hyi (Mauche 1996), and OY Car (Mauche & Raymond 2000). The highinclination system OY Car shows mainly broad emission lines arising due to resonance scattering in the strong disk wind (Mauche & Raymond 2000). The highresolution spectra of the other CVs, obtained by EUVE and Chandra observations, show numerous broad absorption and emissionlike details. These spectra are similar to soft Xray spectra of supersoft Xray sources (Lanz et al. 2005; Rauch et al. 2010) and also have to be modeled using stellar modelatmosphere methods. The first attempt to fit the soft Xray spectrum of SS Cyg in outburst (Mauche 2004) by hot LTE modelatmosphere spectra showed the potential for success of this approach (Suleimanov et al. 2013).
With the work at hand, we start to model the various BL models using the stellaratmosphere method and a comparison of the results with the properties of observed soft Xray/EUV spectra of CVs mentioned above. Here we present the model spectra of particular 1D hydrodynamic BL models that were computed recently by Hertfelder et al. (2013).
2. Method
Our work is based on the 1D models of BLs between optically thick accretion disks and WDs that were computed in our previous work (Hertfelder et al. 2013). Models are considered in a cylindrical coordinate system (z, ϕ, R), being axisymmetric (independent of ϕ) and vertically averaged (over zcoordinate). They were computed for a fixed massaccretion rate Ṁ = 1.5 × 10^{8} M_{⊙} yr^{1}, three different values of WD mass, M_{WD} = 0.8, 1, and 1.2 M_{⊙}, and five values of WD angular velocity, ω_{WD} = 0, 0.2, 0.4, 0.6 and 0.8 ω_{K}, where ω_{K} is the Kepler angular velocity at the WD radius R_{WD}. WD radii were calculated using the Nauenberg (1972) relation. An increase of equatorial radii due to rotation was ignored. A simple α prescription for the viscosity ν_{α} was used according to Shakura & Sunyaev (1973), see details in Hertfelder et al. (2013): (1)where a is the sound speed. We note that ω_{K} is taken for the current BL radius, however, the radial extension of model BLs is low and ω_{K} for a given model is almost constant. A relatively low value of the α parameter (α = 0.01) is assumed for all models in order to avoid supersonic radial motions.
For every 1D model we consider the distribution of the following physical parameters along the radial coordinate R: effective temperature T_{eff}(R) (or, equivalently, the total radiated flux F_{0}(R)), surface density Σ_{0}(R), BL halfthickness H_{0}(R), and rotation velocity v_{ϕ}(R). The following steps have to be performed to compute the model emergent spectrum:

Divide the BL model in a number of rings with equal luminosity.

Compute a gray model for each ring along the vertical coordinate z.

Starting from the gray model, compute a model of the each ring using model atmosphere methods together with the local emergent spectrum.

Sum up the local spectra to a total BL spectrum taking the rotation of the rings into account.
This approach is almost identical to the one that was used for the computation of CV accretion disk spectra by many authors (Kriz & Hubeny 1986; Shaviv & Wehrse 1991; Suleymanov 1992; Wade & Hubeny 1998; Nagel et al. 2004). We understand that the local 1D method assumed here is rather crude. But we believe that it gives a reasonable first approximation, which is applicable to comparison with observations. This method is a first step to more sophisticated 2D models.
Accretion disk and BL in the 1D models presented by Hertfelder et al. (2013) are considered as a comprehensive structure, and the boundary between accretion disk and BL is blurred. However, the T_{eff}(R) distributions have a local minimum at R ≈ 1.1R_{WD}. We adopted that radius as the boundary between the disk and the BL and studied the BLs at radii lesser than this boundary. We divided each BL model into a few (5−20) rings, and the particular radius R_{j} of a given ring with number j, for which a local vertical structure and local emergent spectrum are computed, is determined by the condition (2)where and are the radii of the upper and lower ring boundaries, respectively. We note that .
We now describe the other steps of our method in more detail.
2.1. BL vertical structure. Gray atmosphere approach
The approach employed here is based on the numerical method used for modeling of accretion disk structures over the z coordinate as presented by Suleimanov et al. (2007 – see also Kriz & Hubeny 1986; Shaviv & Wehrse 1991; Suleymanov 1992; Wade & Hubeny 1998; Nagel et al. 2004). Some modifications are made and described additionally.
The vertical structure of each BL ring at radius R_{j} is determined by the ring parameters T_{eff}, Σ_{0}, H_{0}, and a set of differential equations. The first one is the hydrostatic equilibrium equation (3)where P = P_{g} + P_{rad} is the total pressure, the sum of gas and radiation pressure, ρ is the matter density, z_{1} is a parameter, the distance from the midplane to the highest point of the vertical model, and z′ = z_{1} − z is the vertical coordinate, which is equal 0 at the highest point of the model.
The second equation is the energyconservation law. The correct local energy generation rate dF/ dz in accretion disks as well as in BLs is not known and a local version of the α approach is often used (see e.g. Suleimanov et al. 2007). Fortunately, local emergent spectra of optically thick accretion disks (and, therefore, the BL models considered here, too) are weakly dependent on details of the energy generation rate, so we took the simplest version (Shakura & Sunyaev 1973): (4)Here F is the integral (bolometric) vertical flux at given height, , is the emergent integral flux. The energy conservation law has the integral (Shakura & Sunyaev 1973) (5)where the boundary condition F(m = 0) = F_{0} is used. Here the Lagrangian coordinate m is determined by the equation (6)and, therefore, (7)We assumed a purely radiative transport of energy in zdirection, therefore, the third equation is the radiation transfer equation. For the gray approach we use the first moment of that equation (8)where κ_{R} is the Rosseland opacity, which is determined as the greater value of electron scattering, σ_{e} = 0.335 cm^{2} g^{1}, and Kramers opacity, κ_{kr} = 5 × 10^{24}ρT^{3.5}. The sum of these values is also relevant (Popham & Narayan 1995).
These equations are solved together with the ideal gas law (9)where n is the total number density of particles. We assume full local thermodynamic equilibrium (LTE). Therefore, the local gas temperature T can be found from the integrated (over frequency) mean intensity J(10)Here we used the following TE relations between radiation energy density ε_{rad}, integral mean intensity J, and radiation pressure: (11)A solar chemical composition is assumed and all ionization states of the 15 most abundant chemical elements are taken into account. The relation between density ρ, total number density n, and electron number density n_{e} is found using Saha’s equation for each considered ionization state.
Fig. 1 Distribution of BL quantities along radius. The data are from the model with M_{WD} = M_{⊙} and ω_{WD} = 0.8 ω_{K}. The position of the WD surface at R_{WD} = 5.48 × 10^{8} cm is shown by the vertical dotted line. Top panel: shown is the effective temperature (solid curve) together with the relative angular velocity (dashed curve), as obtained from the 1D hydrodynamical model. These are the same for the atmosphere models. Middle and bottom panels: Rosseland optical halfthickness, τ_{0}, and geometrical halfthickness, H_{0}, for the modelatmosphere models (solid lines). The distributions obtained from the 1D hydrodynamical model (dotted curves) and the gray model (dashed curves) are also shown. 

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We solve Eqs. (3) and (5−8) from the surface z′ = 0 to the midplane z′ = z_{1} by using a shooting method with the boundary conditions (12)There is one additional parameter unknown at the outset: z_{1}. We find it by the dichotomy method on the range (1−30) H_{0} using an additional boundary condition at the midplane (13)or, equivalently, (14)A comparison of optical depths and halfthicknesses of the 1D and the grayapproach models along z coordinate for one particular BL model (M_{WD} = M_{⊙} and ω_{WD} = 0.8 ω_{K}) is shown in Fig. 1 in the two bottom panels. It is clear that the new halfthickness of the BL model H = z_{1} − z(τ_{R} = 2/3) is approximately twice as large as the thickness of the 1D hydrodynamic model (red dotted line). This is a well known factor (Shakura & Sunyaev 1973) because the halfthickness of 1D models is just the pressure scale height , while the new thickness is given by the height of the photosphere. The effective temperatures are plotted in the top panel of Fig. 1, here the models agree by definition. The gray temperature structure of the hottest ring of the same BL model is shown in Fig. 2 (bottom panel). We note that this ring is a reference model for our illustration of the presented method. The parameters of the rings for all computed BL models with five rings are shown in Tables 1−3. The ring halfthicknesses, H, and the ring Rosseland optical depths, , were computed by using the gray ring models.
Parameters of the rings for the models with parameters M_{WD} = M_{⊙} and ω_{WD} = 0.8, 0.6, 0.4, 0.2 ω_{K}.
Parameters of the rings for the models with parameters ω_{WD} = 0.8 ω_{K} and M_{WD} = 1.2 and 0.8 M_{⊙}.
Parameters of the rings for the model with parameters M_{WD} = 0.8 M_{⊙} and ω_{WD} = 0.6 ω_{K}.
Every computed gray vertical ring model is interpolated to a logarithmically equidistant column density grid with 98 depth points in the range from ~ 10^{7} g cm^{2} to Σ_{0}/ 2. This interpolated ring model is used as a starting model for the calculation of a nongray model and its emergent spectrum.
Fig. 2 Top panel: local spectra of the hottest ring of the same BL model as in Fig. 1 computed with (solid curve) and without (dashed curve) spectral lines taken into account. The blackbody spectrum corresponding to the effective temperature is shown by the dotted curve. Bottom panel: temperature structures of the model with lines (solid curve) and the gray model (dashed curve). The central temperature of the hydrodynamical 1D model is shown by the dotted line. 

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2.2. Boundary layer vertical structure. Model atmosphere approach
The nongray model is determined by Eqs. (3) and (5), rewritten in the form (15)where g_{z} and g_{rad} are defined by Eqs. (3) and (20). Here we again assume that energy is transferred by radiation alone and this is described by the radiation transfer equation at every considered frequency point (16)where the monochromatic optical depth is determined from (17)and the source function can be expressed as (18)Here μ = cosθ is the cosine of angle θ, which is the angle between the radiation transfer direction and the ring normal. Eq. (16) determines the specific intensity I_{ν} at given frequency and μ. The mean intensity J_{ν} and Eddington flux H_{ν} = F_{ν}/ 4π are defined by (19)We took coherent electron scattering σ_{e} into account together with the freefree and boundfree opacities k_{ν} of all ions of the 15 most abundant elements using opacities from Verner & Yakovlev (1995) and Verner et al. (1996). Line blanketing was taken into account using ~ 25 000 spectral lines from the CHIANTI, Version 3.0, atomic database (Dere et al. 1997).
Fig. 3 Top panel: sound speed (dashed curve) and wind velocity (solid curve) distributions along depth in the ring model shown in Fig. 2. Middle panel: distributions of gravity (dashed curve) and radiative acceleration (solid curve) along depth. The position of the sonic point is marked by the dotted vertical line. Bottom panel: depths where the emergent spectrum forms (τ_{λ} = 1). The position of the sonic point is marked by the dashed horizontal line. 

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We solve the radiation transfer equation (16) at three values of  μ  using the short characteristics method (Olson & Kunasz 1987). The adopted μ values correspond to ChebyshevGauss quadrature abscissas to accurately compute the integrals in (19).
Radiative acceleration g_{rad} is determined by (20)A correct model has to satisfy the energy balance equation which can be written in two forms: (21)and (22)Of course, the initial model does not satisfy the energy balance equation, having the relative flux error (23)and flux derivative error (24)at each depth. It is possible to find corrections to the temperature ΔT(m) using two methods modified for a nonconstant bolometric flux over the ring depth (Suleymanov 1992): the integral Λiteration together with a surface temperature correction for the optically thin parts of the atmosphere (25)where α_{ν} = σ_{e}/ (k_{ν} + σ_{e}), and Λ_{ν,diag} is the diagonal matrix element of the Λoperator, and the AvrettKrook flux correction based on the relative flux error ε_{H}(m) for the optically thick parts of the ring model (26)Both methods were described in detail by Kurucz (1970).
Then we find new values of the gas pressure using new g_{rad} and recalculate the densities. Subsequently, opacities are recalculated and the radiation transfer equations at all frequencies are resolved and a new temperature structure is computed. Then we recalculate the geometrical depth scale z′(m) using the new densities. This procedure is performed up to convergence using our version of the computer code ATLAS (Kurucz 1970) that was modified to deal with high temperatures (Ibragimov et al. 2003; Suleimanov & Werner 2007; Suleimanov et al. 2013). We assumed LTE and accounted for pressure ionization effects using the occupation probability formalism (Hummer & Mihalas 1988), as described by Hubeny et al. (1994).
Because of the significant effect of spectral lines on the radiation force g_{rad}, the atmosphere modeling approach is divided into two steps. In the first step we compute a continuum model without spectral lines, which provides the radiation force g_{rad}<g_{z} at all BL ring depths. The distributions of basic BL quantities for continuum ring models are presented in Fig. 1 by solid curves. Hydrostatic ring models cannot be computed when spectral lines are included, because then the radiation force g_{rad} becomes higher than the gravity g_{z} in the upper, optically thin (in continuum) layers of the models. Therefore, a linedriven wind has to arise from the BL surface, which is similar to the linedriven winds from hot stars (Castor et al. 1975; Kudritzki & Puls 2000). This kind of wind also exists for CV accretion disks (Drew & Verbunt 1985; Prinja & Rosen 1995; Proga et al. 1998).
2.2.1. Linedriven wind approach
Here, we are not interested in the properties of the wind itself, because it probably effects only the strongest spectral lines but not the continuum or moderate and weak spectral lines. In previous works (Ibragimov et al. 2003; Suleimanov et al. 2013), where hot WD model atmospheres were computed, the gas pressure was artificially fixed to 10% of the total pressure at the upper layers, where g_{rad}>g_{z}. The condition P_{gas} = 0.1P was chosen because it was approximately correct at those atmospheric layers, where g_{rad} ≈ g_{z}. This condition physically means that we assume some artificial wind velocity law that satisfies the imposed condition. Unfortunately, this simple approach is not working for the BL case because gravity g_{z} depends on the vertical coordinate z (it is constant for ordinary stellar atmospheres). Therefore, we develop another simple hydrodynamic approach which takes into consideration an expansion of the upper ring layers. This approach is not selfconsistent as we do not include effects of motion in the radiation transfer. Thus we consider a hydrodynamical model for the ring upper layers but the radiation pressure force governing this expansion is computed for a formally static medium. This approach allows computing the model emergent spectra but the obtained wind properties such as the local massloss rate and windvelocity distributions are not completely correct and cannot be considered as reliable results.
The considered approach is based on Euler’s equation (27)where ṁ is the local massloss rate ([ṁ]= g s^{1} cm^{2}) and v is the gas velocity at given z. This equation can be rewritten as (28)where a is the sound speed connected with gas pressure and matter density by (29)Here we used the continuity equation for the plane 1D motion (30)It is well known (see, e.g. Mihalas 1978) that a correct solution of Eq. (28) has to pass through a singular point with  v  = a, where the left and the right side of the equation vanish simultaneously. Therefore, these two conditions must be fulfilled in the singular point: (31)and (32)or (33)We suggest the following scheme to compute a model of the BL ring with a linedriven wind. We take the model computed without lines as an initial model and solve the radiationtransfer equation with spectral lines taken into account. As a result we obtain that the radiative acceleration g_{rad} is greater than the current gravity at the upper layers. At this stage we have the gas pressure P_{g}(m) and the gas density ρ(m) distributions from the continuum model, and the radiative acceleration g_{rad}(m) and the gas temperature T(m) distributions after the first temperaturecorrection iteration. The geometrical depth scale z(m) and corresponding gravity g_{z} are connected with the gas density of the initial (continuum) model. Using these distributions we can find the soundspeed distribution a(m) and its derivative da(m)/dz, and then find the depth m_{c} where Eq. (32) is satisfied. We know at this depth the wind velocity must be equal to the sound speed a(m_{c}). Therefore, we can evaluate the local massloss rate (34)using the continuity Eq. (30). Here we have taken into account that the correct gas density ρ_{w} in the singular point for the model with the wind must be half of that in the hydrostatic model ρ_{st}, see Eq. (27) (35)Using this massloss rate ṁ we solve two ordinary differential equations (36)and (37)where P_{w} = ṁv + ρa^{2}, on a new fine and equidistant geometrical depth grid z (50 000 points) from the midplane (z = 0) up to the surface (m′ = Σ_{0}/ 2) with the boundary conditions v = 0, m′ = 0, and P_{w}(0), ρ(0) = P_{w}(0) /a(0)^{2} at the midplane, using the shooting method. The current gas density is calculated using the current P_{w}(38)The plus sign is used for wind velocities below the sonic point and the minus sign in the supersonic part of the model. The wind velocity is found using the continuity Eq. (30). The necessary current values of the g_{rad}(m′) and da^{2}(m′)/dz = ρd a^{2}(m′)/d m′ are found using a spline interpolation in the dependencies g_{rad}(m) and da^{2}(m)/dm known from the previous iteration, taking m = Σ_{0}/ 2 − m′.
Usually, the necessary conditions Eqs. (28) and (32) are not satisfied at the sonic point for the first tried initial condition for P_{w}(0). Therefore, we find the necessary P_{w}(0) with the dichotomy method. Usually, the correct initial value of the gas pressure at the midplane differs by a few percent from the gas pressure at the midplane found in the previous iteration. We adopt the two following numerical convergence criteria at the singular point After obtaining the solution, all quantities are interpolated on the model atmosphere column density grid m (98 points) which is used for the radiation transfer solution. Therefore, we have a new distribution of the gas pressure P_{g} = ρa^{2} with the linedriven wind taken into account which is in accordance with the radiative acceleration g_{rad}. Then we find a new gas density distribution using the new P_{g} and the temperature distribution obtained after the temperaturecorrection iteration. For this aim the ideal gas Eq. (9) is used together with charge and number density conservation laws. Then the radiation transfer is resolved and a new temperature correction is performed. The above described procedure to find a new gaspressure distribution is repeated with a new radiativeacceleration distribution. There is only one difference. The local massloss rate is now calculated (see Eq. (34)) without division by 2 because the current model takes into consideration the linedriven wind.
This iteration scheme is converging and it is stable. As a result we get a selfconsistent model of the BL ring and its emergent spectrum. Results of calculations for the reference ring model are presented in Figs. 2 and 3. The emergent spectrum (Fig. 2, top panel) is similar to hot stellar atmospheres with similar effective temperatures (see e.g. Suleimanov et al. 2013) and shows the same absorption edges as the continuum spectrum together with a forest of spectral absorption lines. The temperature structure (Fig. 2, bottom panel) differs significantly from the gray temperature distribution mainly because of lineblanketing effects. The final model’s wind velocity law (Fig. 3, top panel) and the radiation force g_{rad} distribution (Fig. 3, middle panel) are shown together with the sound speed a and gravity g_{z} distributions. The sonic point is seen to be located at the upper ring layers (m ≈ 10^{5} g cm^{2}) and the corresponding massloss rate is sufficiently low (ṁ ≈ 10^{5}−10^{6} g s^{1} cm^{2}). Formally, the sonic point is above the formation depths of even the strongest spectral lines (Fig. 3, bottom panel) and, therefore, the wind does not affect the spectrum.
2.3. Integral BL spectra
When the computation of all ring model spectra is finished, we integrate them over the BL radial coordinate with taken Doppler broadening due to ring rotation into account (39)where i is the inclination angle of the BL relative to the line of sight, I_{λ}(cosi) is the local specific intensity in the direction of the line of sight, N_{R} is the number of considered BL rings (5 or 20), and N_{ϕ} = 100 is a number of considered ring sectors. Here λ′ is the Doppler shifted wavelength: (40)The azimuthal velocity distribution v_{ϕ}(R) is taken from the 1D BL model. We note that integration is only performed over the visible BL part. We assume that we see only a quarter of the total BL surface, which is situated in front of the WD on the visible part of the disk, and that the visible part does not depend on the inclination angle. The change of the projection area of the BL’s visible part is proportional to cosi. Specific intensities for every ring model are computed using Eq. (16) for six angles relative to the normal: 0, 15, 30, 45, 60 and 75 degrees. Therefore, integral spectra can be computed for these inclination angles relative to the line of sight. We ignored any relativistic effects because they are low.
Fig. 4 Spectra of the BL model M = M_{⊙}, ω_{WD} = 0.8 ω_{K} for three inclination angles relative to the line of sight. The corresponding blackbody approximations are shown by dashed curves. 

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Fig. 5 Detail of the integral emergent fluxes of the same BL model as in Fig. 4 computed for i = 0° (rotation does not affect the spectrum, thin black curve), and 15° (thick red curve). 

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Fig. 6 Comparison of the spectra of the same BL model (see Fig. 4) computed using five (dotted curve) and twenty (solid curve) rings. 

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3. Results
Using the method presented above, we computed spectra of several BL models. Examples of the integral reference BL model spectra for three angles are shown in Fig. 4. These spectra are calculated using the BL model partitioned in 20 rings. The spectra show broad absorption and quasiemission features. The absorption features arise due to rotational broadening and blending as a result of many absorption lines at this place of the spectrum. The quasiemission features are continuum sections with a low amount of absorption lines. These statements are illustrated by Fig. 5, where a narrow section of the integral spectra computed for i = 0° and 15° are shown. In Fig. 4 the spectra computed in blackbody approximation are also shown. We focus attention on the fact that the atmospheremodel intensities compared to the blackbody spectra are different for low and high inclination angles. This is the result of different limbdarkening laws for the ring atmospheres (the intensity is mainly concentrated along the normal) compared to the isotropic blackbody radiation.
We investigated the importance of the number of rings to describe the model spectra. In addition to the spectrum of the BL reference model presented above, we computed the spectrum of the same model using five rings. A comparison of both spectra is shown in Fig. 6. The difference is significant at wavelengths longer than 120−130 Å, but the spectra are almost identical at the shorter wavelengths. Our work was particularly motivated by the Chandra Xray spectrum of SS Cyg in outburst (Mauche 2004). The grating spectra cover the wavelength range 10−130 Å, therefore, to model the observed BL spectra five rings are sufficient. Moreover, even the hottest ring spectrum alone can approximate the integral BL spectrum with a relatively good accuracy (see Fig. 7). All BL spectra of the two grids described above were computed using five rings.
Fig. 7 Comparison of the BL model spectrum from Fig. 4 (solid curve) with the spectrum of the hottest ring (dotted curve) normalized to the flux at 100 Å. 

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Fig. 8 Effective temperature distributions (top panel, the temperatures of the considered rings are marked by dots) and model spectra (bottom panel) for the models with different angular velocities (0.2, 0.4, 0.6 and 0.8 ω_{K}) and fixed WD mass (M_{WD} = M_{⊙}). 

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Fig. 9 Spectra of the same models as in Fig. 8 in the observed wavelength range. The spectra in the top panel are normalized to the ω_{WD} = 0.6 ω_{K} model spectrum at 100 Å. 

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Fig. 10 Comparison of BL model spectra for three WD masses (0.8, 1, 1.2 M_{⊙}) and fixed relative angular velocity (ω_{WD} = 0.8 ω_{K}). The spectra in the top panel are normalized to the M_{WD} = M_{⊙} model flux at 100 Å. 

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It is important to explore the dependence of the BL model spectra on the BL parameters. For this purpose, we computed spectra of two model grids. In the first one we fixed the relative WD angular velocity ω_{WD} = 0.8 ω_{K} and investigated how the spectra depend on the WD mass M_{WD}. In the second set we fixed the WD mass M_{WD} = M_{⊙} and varied the relative WD angular velocity. The obtained results are described below.
A comparison of the computed integral BL spectra for various parameters is shown in Figs. 8−10. The basic differences of the presented spectra are determined by differences in the BL model effective temperatures (see e.g. Fig. 8, top panel). The radial widths of all the computed models are similar (see the same figure and Hertfelder et al. 2013), but the luminosities can change more significantly. So, the BL effective temperatures directly connect with the BL luminosities, which are defined by mass accretion rate Ṁ, WD mass, and relative WD angular velocity ω_{WD}/ω_{K} (Kluźniak 1987; Kley 1991): (41)All the models considered here have a fixed mass accretion rate. Therefore, luminosities and effective temperatures of these models only depend on WD masses and relative angular velocities. Both of them increase with the WD mass and decrease with the relative angular velocity. Higher BL effective temperatures have to correspond to harder spectra and higher ionization degree of the matter.
Our computations confirm these qualitative conclusions. Emergent spectra of the BL models around a solar mass WD with various relative angular velocities are shown in Fig. 8. The effective temperatures increase with decreasing WD relative angular velocity, and the emergent spectra become harder. As a result, the photoabsorption edges and the intensities of quasiabsorption and quasiemission features are changing. In particular, the position of a notable spectral shortwavelength rolloff shifts from 60 Å in the spectrum of BL model around the fastest rotating WD up to 25 Å in the spectrum of the BL model around the slowest rotating WD. The same spectra in the 15−130 Å wavelength band on a linear flux scale are shown in Fig. 9. The spectra, which are normalized at 100 Å, are also shown in the top panel of the same Figure to emphasize the changes in the spectral shapes.
We note that some ring models (near the minimum of the surface density distribution) of the BL model around the nonrotating WD (M_{WD} = M_{⊙}, ω_{WD} = 0) did not converge and a corresponding integral BL model spectrum was not computed. The reason is that these rings are effectively optically thin () over a wide wavelength range. We can see that the average Rosseland opacities for the hottest rings for the BL models around slowly rotating WDs are close to pure electron scattering and the input of the “true” opacity k_{kr} is negligible, because τ_{0} ≈ σ_{e}Σ_{0}/ 2 (see Table 1). Possibly, the similar effect may exist for other BL models around nonrotating WDs. It could be less important for the BL around WDs with lower masses and could be reduced for lesser mass accretion rates.
The difference between BL spectra computed for the fixed WD relative angular velocity but various WD masses are qualitatively similar (Fig. 10). The spectrum of the BL model around the heaviest WD is harder than the spectra of BL models around less heavy WDs.
4. Comparison with observations
As mentioned in the introduction, there are only four dwarf novae with observed soft Xray/EUV radiation in outbursts, which can be associated with optically thick BLs. One of them is SS Cyg (Mauche 2004; Suleimanov et al. 2013). Fitting the spectrum gave effective temperature estimations of 190 (250) kK and a bolometric BL luminosity 18(5) × 10^{33} erg s^{1} for an assumed distance of 160 pc. The bolometric disk luminosity of SS Cyg at the peak of the outburst, L_{D} ≈ 10^{35} erg s^{1}, was estimated by Mauche (2004) using the same distance and the data published by Polidan & Holberg (1984). We note that the soft Xray spectrum of SS Cyg was obtained by Chandra at the outburst peak, too. The corresponding mass accretion rate Ṁ ≈ 1.5 × 10^{8} M_{⊙}/yr, which was used for the computation of all the BL models in this work, was obtained using this luminosity for the adopted WD parameters, M_{WD} = M_{⊙} and R_{WD} = 5.5 × 10^{8} cm. The corresponding ratio of BL luminosity L_{BL} to disk luminosity L_{D} is 0.18 (0.05). These values correspond to the fit of the spectrum by a hot stellar atmosphere that is close to Eddington limit (with log g = 6.2, Suleimanov et al. 2013), and the values in brackets were obtained using blackbody fits with broad absorption lines (Mauche 2004). The recent determination of SS Cyg’s distance by Nelan & Bond (2013) gave a lesser value, ~110 pc instead of ~160 pc. It reduces the disk and the BL luminosities mentioned before, but does not change their ratio. The low value of L_{BL}/L_{D} supports the hypothesis about a fast WD rotation in SS Cyg (Mauche 2004). In the ultraviolet spectrum of this CV in quiescence the absorption lines, which are in principle available to determine the WD’s projected rotation velocity, were not found (Sion et al. 2010). The most probable reason for this finding are strong emission lines in the UV spectrum of SS Cyg in quiescence (Long et al. 2005), which could mask absorption lines in the WD spectrum. But it is necessary to keep in mind that the BL luminosity estimations are very model dependent and have large uncertainties.
Parameters of the strongest emission lines, preliminarily identified in the subtracted spectrum.
We compare our BL model spectra to the observed Chandra spectrum of SS Cyg (Mauche 2004), using the well known normalization for converting the calculated spectra to the spectrum at Earth (42)where d is the distance. A relatively good fit was obtained for the model spectrum of a BL with parameters M_{WD} = M_{⊙},ω_{WD} = 0.8 ω_{K} for the distance d ≈ 107 pc and the absorption by the neutral interstellar medium with N_{H} ≈ 7 × 10^{19} cm^{2} (Fig. 11). We see that the model spectrum describes some prominent features in the observed spectrum, but there are also some additional features. The subtracted (observed minus BL model) spectrum (Fig. 11, bottom panel) is very similar to a spectrum of an optically thin hot plasma with a strong emission continuum of NeVII and numerous broad emission lines. Part of them are identified (see Fig. 11 and Table 4) as spectral lines with great gf values. It is most probable that the difference spectrum is due to an optically thin outflow, which is projected on the cool WD and the accretion disc, see Fig. 12. We also show in the bottom of Fig. 11 the absorbed (with the same N_{H}) spectrum of the homogeneous slab computed in LTE approximation with temperature T = 160 kK, density ρ = 10^{7} g cm^{3}, and surface density Σ_{W} = 6 × 10^{2} g cm^{2}. The corresponding geometrical thickness of the slab is 6 × 10^{5} cm. The spectrum is computed by the short characteristics method and broadened using a Gauss function with σ = 0.2 Å. This model slab spectrum cannot fit all the emission details correctly, because the real outflow has to be strongly inhomogeneous, and, probably, the LTE approximation is not correct there. But it reproduces the general shape of the subtracted spectrum supporting our hypothesis. We found that the emission area of this slab is approximately four times greater than the visible model BL area. If we assume that the obtained density of the slab is correct for the averaged density of the outflow, we can estimate a total mass loss rate of Ṁ_{W} ~ ρv_{esc}S_{BL} ~ 10^{7} g cm^{3} × ~ 10^{8} cm s^{1} × ~ 10^{16}−10^{17} cm^{2}~ 10^{17}−10^{18} g s^{1}. This guess is very crude, but, nevertheless, it is comparable with the mass accretion rate, and, therefore, future BL models have to be computed with mass loss taken into account.
Fig. 11 Top panel: comparison of the observed SS Cyg soft Xray spectrum (solid curve) with the BL model spectrum for a 1 M_{⊙} WD and relative angular velocity ω_{WD} = 0.8 ω_{K} (dashed curve). Bottom panel: subtracted (observed minus BL model) spectrum (solid curve) together with the homogeneous slab model spectrum (dashed curve). The emission continuum of Ne VII below 60 Å and the strongest identified emission lines are marked. 

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Fig. 12 Scheme of the BL outflow. 

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Fig. 13 Comparison of the BL model spectra for a 0.8 M_{⊙} WD and relative angular velocity ω_{WD} = 0.6 ω_{K} (dashed curve) and for a 1 M_{⊙} WD and relative angular velocity ω_{WD} = 0.8 ω_{K} (solid curve). The flux of the latter spectrum is multiplied by 2.5. 

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However, we have to point out that a recent estimation gave a lesser WD mass in SS Cyg, M_{WD} = 0.81 ± 0.18 M_{⊙} (Bitner et al. 2007) instead of 1 M_{⊙}. We may expect that some BL model around a slower rotating WD with M_{WD} = 0.8 M_{⊙} can have similar effective temperatures (and a similar spectrum) as the considered BL model with M_{WD} = M_{⊙} and ω_{WD} = 0.8ω_{K}. Indeed, we find that the spectra of BL models with parameters M_{WD} = 0.8M_{⊙},ω_{WD} = 0.6 ω_{K} and M_{WD} = M_{⊙},ω_{WD} = 0.8 ω_{K} are very similar (see Fig. 13). Therefore the spectrum of the BL model with parameters M_{WD} = 0.8 M_{⊙},ω_{WD} ≈ 0.6 ω_{K} also has to fit the observed SS Cyg soft Xray spectrum in the outburst with d ≈ 170 pc. The ratios L_{BL}/L_{D} are 0.04 and 0.16 for these BL models, respectively, and they are close to the estimations obtained by Mauche (2004) and Suleimanov et al. (2013). Therefore, the BL model spectra computed here can fit the soft Xray spectrum of SS Cyg in outburst.
The second bright dwarf nova that exhibited a prominent soft Xray/EUV flux during outburst is U Gem (Cordova et al. 1984; Long et al. 1996). But the properties of this emission differ significantly from the case of SS Cyg. A blackbody fit gives a temperature about 140 kK and a BL luminosity of about 4 × 10^{34} erg s^{1} (for the distance 90 pc, Marsh et al. 1990) with the corresponding ratio L_{BL}/L_{D} ~ 0.5 (Long et al. 1996). The earlier Xray observations performed by the Einstein observatory gave similar relations between the blackbody temperatures and the observed fluxes of these CVs’ soft Xray spectra (Cordova et al. 1980a,b, 1984).
The estimated rotation velocity of the WD in this CV is very low (< 100 km s^{1}, Sion et al. 1994) and the derived relative BL luminosity is in accordance with the expected relative BL luminosity (see Eq. (41)). But all 1D BL models including ours predict higher effective temperatures for this luminous BL, at least 200−300 kK, in contradiction with the observed temperature. This means that the BL emitting area in U Gem is about 5−10 times greater than in SS Cyg, i.e. the BL width can be tens of percent of the WD radius instead of a few percent. Therefore, a new BL model is necessary to explain this fact.
A correct theory of optically thick BLs has to explain the observational properties of both dwarf novae. A possible reason could be the 2D nature of BLs around slowly rotating WDs. For example, the BL matter can spread over the WD surface increasing the radiating area (Kley 1991; Fisker & Balsara 2005; Balsara et al. 2009). Another possible physical mechanism is deposition of a significant part of BL energy deep into the WD (Kippenhahn & Thomas 1978; Godon et al. 1995), as it was also considered for T Tau type stars (Bertout & Regev 1992; Regev & Bertout 1995), with subsequent radiation of this energy by a significant part of the WD surface above and below the equatorial plane. An additional possibility was suggested by Idan & Shaviv (1996), who proposed that a significant part of the energy released in the BL might be carried away by some outflow (wind).
The EUV spectrum of U Gem in outburst is very similar to the soft Xray spectrum of SS Cyg in outburst. It also shows numerous broad absorption and emissionlike features, but a few strong emission lines are also detected (Long et al. 1996). The latter probably arise in a strong BL wind and they are visible due to a higher inclination of U Gem’s orbital plane relative to the line of sight (i ≈ 70°, Zhang & Robinson 1987) in comparison with the inclination angle of SS Cyg (i ≈ 50°, Bitner et al. 2007). The even more inclined SU UMatype dwarf nova OY Car (i ≈ 83°, Littlefair et al. 2008) shows an EUV spectrum dominated by prominent emission lines in superoutburst as presented by Mauche & Raymond (2000). These authors evaluated the wind massloss rate (≤ 10^{10} M_{⊙} yr^{1}) and argued that this value cannot be explained by a linedriven disk wind. It is possible that a linedriven BL wind as presented in our calculations can help to resolve this problem. We suggest that radiationdriven winds are much more powerful for BLs around slowly rotating WDs because of much higher effective temperatures at the same emitting area (see, e.g., Table 2). Therefore, such winds could be optically thick like the winds of WolfRayet stars (see, e.g. Nugis & Lamers 2002; Gräfener & Hamann 2005), and the visible BL photospheres could have much greater emitting areas in comparison with the expected BL model sizes. This would lead to significant reduction of the averaged BL effective temperature while saving the bolometric luminosity. This hypothesis can explain the observed properties of the BL in U Gem. We suggest to call this kind of BLs “photospheric radius expanded” (PRE) BLs. A detailed BL wind investigation would be necessary to test these ideas.
5. Conclusions
In this paper we presented the first attempt to compute the soft Xray/EUV spectra of optically thick BL models in CVs using the model stellaratmosphere method. We used the 1D hydrodynamic BL models calculated by Hertfelder et al. (2013) for three WD masses (0.8, 1, and 1.2 M_{⊙}) and various values of relative angular velocity (0, 0.2, 0.4, 0.6, and 0.8 ω_{K}). Two other parameters, the massaccretion rate Ṁ = 1.5 × 10^{8} M_{⊙} yr^{1} and the viscosity parameter α = 0.01 were fixed. Every BL model is divided into a few rings, and every ring model along the zdirection is computed using the modelatmosphere method. The total BL model spectra are summed over the ring spectra with Doppler effect taken into account.
The effective temperatures of the considered BL model rings range from 100 kK to 400 kK. At these conditions hydrogen and helium are almost fully ionized and highlycharged ions of heavy elements such as carbon, oxygen, neon, magnesium, and silicon determine the opacities and the shape of the model emergent spectra.The local ring spectra with numerous absorption lines of these chemical elements together with absorption edges are smeared by the fast ring rotation and the final model BL spectra show relatively broad absorption and emissionlike features, which cannot be identified with any individual absorption or emission lines. The absorptionlike features arise at the spectral parts with high number densities of strong absorption lines and/or near blue sides of absorption edges. The emissionlike features arise at spectral parts with a low number densities of absorption lines and/or near red sides of absorption edges. An additional smoothing of the spectra due to a finite spectral resolution of Xray instruments has to be also considered to form these prominent features. The predominance of spectral lines in the BL opacities leads to strong BL linedriven winds, which manifest themselves in EUV observations. We developed a simple method to treat approximately the influence of a linedriven wind on the BL model structure. This approach is not selfconsistent and it is not sufficiently correct for an investigation of the wind properties, but it offers the possibility to compute the model emergent spectra.
The observed soft Xray and EUV spectra of the dwarf novae SS Cyg and U Gem in outburst are very similar to the computed model BL spectra and exhibit numerous emission and absorptionlike features (Mauche 2004; Long et al. 1996). Moreover, we reveal that the soft Xray Chandra spectrum of SS Cyg in outburst can be fitted satisfactorily by the spectrum of our BL models with M_{WD} = M_{⊙}, ω_{WD} = 0.8 ω_{K}, and M_{WD} = 0.8 M_{⊙}, ω_{WD} = 0.6 ω_{K}. with the interstellar absorption parameter N_{H} ≈ 7 × 10^{19} cm^{2}, and the distances ≈107 and 170 pc correspondingly. The ratio of the observed BL luminosity to the disk luminosity of SS Cyg in outburst is consistent with the model parameters. The subtracted spectrum is a spectrum of a hot optically thin plasma spectrum and could be associated with the outflow from the BL surface with the mass loss rate ~ 10^{17}−10^{18} g s^{1}, which is comparable with the assumed mass accretion rate. Therefore, future BL models have to be computed with the mass loss taken into consideration.
On the other hand, the observed EUV flux properties of U Gem in outburst are contradicting our model predictions. We suggest that a powerful radiationdriven wind from U Gem’s BL could form an extended photospheric BL surface reducing the averaged effective temperature but saving the bolometric luminosity. The properties of this “photospheric radius expanded” BL could be very similar to the observed U Gem BL properties. Additional extended BL radiationdriven wind investigations are necessary to proof this hypothesis.
Acknowledgments
We thank Chris Mauche, who put the Chandra Xray spectrum of SS Cyg at our disposal and who helped us to fit it by our models, and the anonymous referee for numerous helpful remarks. V.S. thanks DFG for financial support (grant SFB/Transregio 7 “Gravitational Wave Astronomy”) and Russian Foundation of Fundamental Research (grant 120297006rpovolzhea). M.H. received financial support from the German National Academic Foundation (Studienstiftung des deutschen Volkes).
References
 Balbus, S. A., & Hawley, J. F. 1991, ApJ, 376, 214 [NASA ADS] [CrossRef] [Google Scholar]
 Balsara, D. S., Fisker, J. L., Godon, P., & Sion, E. M. 2009, ApJ, 702, 1536 [NASA ADS] [CrossRef] [Google Scholar]
 Baskill, D. S., Wheatley, P. J., & Osborne, J. P. 2005, MNRAS, 357, 626 [NASA ADS] [CrossRef] [Google Scholar]
 Belyaev, M. A., & Rafikov, R. R. 2012, ApJ, 752, 115 [NASA ADS] [CrossRef] [Google Scholar]
 Belyaev, M. A., Rafikov, R. R., & Stone, J. M. 2012, ApJ, 760, 22 [NASA ADS] [CrossRef] [Google Scholar]
 Belyaev, M. A., Rafikov, R. R., & Stone, J. M. 2013, ApJ, 770, 67 [NASA ADS] [CrossRef] [Google Scholar]
 Bertout, C., & Regev, O. 1992, ApJ, 399, L163 [NASA ADS] [CrossRef] [Google Scholar]
 Bitner, M. A., Robinson, E. L., & Behr, B. B. 2007, ApJ, 662, 564 [NASA ADS] [CrossRef] [Google Scholar]
 Castor, J. I., Abbott, D. C., & Klein, R. I. 1975, ApJ, 195, 157 [NASA ADS] [CrossRef] [Google Scholar]
 Chandrasekhar, S. 1960, Proc. National Academy of Science, 46, 253 [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
 Collins, T. J. B., Helfer, H. L., & Van Horn, H. M. 1998, ApJ, 508, L159 [NASA ADS] [CrossRef] [Google Scholar]
 Collins, T. J. B., Helfer, H. L., & Van Horn, H. M. 2000a, ApJ, 534, 934 [NASA ADS] [CrossRef] [Google Scholar]
 Collins, T. J. B., Helfer, H. L., & Van Horn, H. M. 2000b, ApJ, 534, 944 [NASA ADS] [CrossRef] [Google Scholar]
 Cordova, F. A., Chester, T. J., Tuohy, I. R., & Garmire, G. P. 1980a, ApJ, 235, 163 [NASA ADS] [CrossRef] [Google Scholar]
 Cordova, F. A., Nugent, J. J., Klein, S. R., & Garmire, G. P. 1980b, MNRAS, 190, 87 [NASA ADS] [CrossRef] [Google Scholar]
 Cordova, F. A., Chester, T. J., Mason, K. O., Kahn, S. M., & Garmire, G. P. 1984, ApJ, 278, 739 [NASA ADS] [CrossRef] [Google Scholar]
 Dere, K. P., Landi, E., Mason, H. E., Monsignori Fossi, B. C., & Young, P. R. 1997, A&AS, 125, 149 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Done, C., Osborne, J. P., & Beardmore, A. P. 1995, MNRAS, 276, 483 [NASA ADS] [CrossRef] [Google Scholar]
 Drew, J., & Verbunt, F. 1985, MNRAS, 213, 191 [NASA ADS] [CrossRef] [Google Scholar]
 Fisker, J. L., & Balsara, D. S. 2005, ApJ, 635, L69 [NASA ADS] [CrossRef] [Google Scholar]
 Frank, J., King, A., & Raine, D. J. 2002, Accretion Power in Astrophysics: 3rd edn. (Cambridge: Cambridge University Press) [Google Scholar]
 Gilfanov, M., Revnivtsev, M., & Molkov, S. 2003, A&A, 410, 217 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Godon, P. 1995a, MNRAS, 277, 157 [NASA ADS] [Google Scholar]
 Godon, P. 1995b, MNRAS, 274, 61 [NASA ADS] [CrossRef] [Google Scholar]
 Godon, P. 1997, ApJ, 483, 882 [NASA ADS] [CrossRef] [Google Scholar]
 Godon, P., & Sion, E. M. 2011, PASP, 123, 903 [NASA ADS] [CrossRef] [Google Scholar]
 Godon, P., Regev, O., & Shaviv, G. 1995, MNRAS, 275, 1093 [NASA ADS] [CrossRef] [Google Scholar]
 Godon, P., Sion, E. M., Levay, K., et al. 2012, ApJS, 203, 29 [NASA ADS] [CrossRef] [Google Scholar]
 Gräfener, G., & Hamann, W.R. 2005, A&A, 432, 633 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Hertfelder, M., Kley, W., Suleimanov, V., & Werner, K. 2013, A&A, 560, A56 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Hubeny, I., Hummer, D., & Lanz, T. 1994, A&A, 282, 151 [NASA ADS] [Google Scholar]
 Hummer, D., & Mihalas, D. 1988, ApJ, 331, 794 [NASA ADS] [CrossRef] [Google Scholar]
 Ibragimov, A. A., Suleimanov, V. F., Vikhlinin, A., & Sakhibullin, N. A. 2003, Astron. Rep., 47, 186 [NASA ADS] [CrossRef] [Google Scholar]
 Idan, I., & Shaviv, G. 1996, MNRAS, 281, 604 [NASA ADS] [CrossRef] [Google Scholar]
 Inogamov, N. A., & Sunyaev, R. A. 1999, Astron. Lett., 25, 269 [NASA ADS] [Google Scholar]
 Inogamov, N. A., & Sunyaev, R. A. 2010, Astron. Lett., 36, 848 [NASA ADS] [CrossRef] [Google Scholar]
 Jones, M. H., & Watson, M. G. 1992, MNRAS, 257, 633 [NASA ADS] [CrossRef] [Google Scholar]
 Kato, S., & Inagaki, S. 1994, PASJ, 46, 289 [NASA ADS] [Google Scholar]
 Kippenhahn, R., & Thomas, H.C. 1978, A&A, 63, 265 [NASA ADS] [Google Scholar]
 Kley, W. 1989, A&A, 222, 141 [NASA ADS] [Google Scholar]
 Kley, W. 1991, A&A, 247, 95 [NASA ADS] [Google Scholar]
 Kley, W., & Hensler, G. 1987, A&A, 172, 124 [NASA ADS] [Google Scholar]
 Kluźniak, W. 1987, PhD thesis, Stanford University [Google Scholar]
 Kriz, S., & Hubeny, I. 1986, Bull. Astr. Inst. Czechosl., 37, 129 [NASA ADS] [Google Scholar]
 Kudritzki, R.P., & Puls, J. 2000, ARA&A, 38, 613 [NASA ADS] [CrossRef] [Google Scholar]
 Kurucz, R. L. 1970, SAO Special Report, 309 [Google Scholar]
 Lanz, T., Telis, G. A., Audard, M., et al. 2005, ApJ, 619, 517 [NASA ADS] [CrossRef] [Google Scholar]
 Lewin, W. H. G., van Paradijs, J., & Taam, R. E. 1993, Space Sci. Rev., 62, 223 [NASA ADS] [CrossRef] [Google Scholar]
 Littlefair, S. P., Dhillon, V. S., Marsh, T. R., et al. 2008, MNRAS, 388, 1582 [NASA ADS] [CrossRef] [Google Scholar]
 Long, K. S., Mauche, C. W., Raymond, J. C., Szkody, P., & Mattei, J. A. 1996, ApJ, 469, 841 [NASA ADS] [CrossRef] [Google Scholar]
 Long, K. S., Froning, C. S., Knigge, C., et al. 2005, ApJ, 630, 511 [NASA ADS] [CrossRef] [Google Scholar]
 Long, K. S., Brammer, G., & Froning, C. S. 2006, ApJ, 648, 541 [NASA ADS] [CrossRef] [Google Scholar]
 LyndenBell, D., & Pringle, J. E. 1974, MNRAS, 168, 603 [NASA ADS] [CrossRef] [Google Scholar]
 Marsh, T. R., Horne, K., Schlegel, E. M., Honeycutt, R. K., & Kaitchuck, R. H. 1990, ApJ, 364, 637 [NASA ADS] [CrossRef] [Google Scholar]
 Mauche, C. W. 1996, in Cataclysmic Variables and Related Objects, eds. A. Evans & J. H. Wood, Astrophys. IAU Colloq., 158, Space Sci. Library, 208, 243 [Google Scholar]
 Mauche, C. W. 2004, ApJ, 610, 422 [NASA ADS] [CrossRef] [Google Scholar]
 Mauche, C. W., & Raymond, J. C. 2000, ApJ, 541, 924 [NASA ADS] [CrossRef] [Google Scholar]
 Mauche, C. W., Raymond, J. C., & Mattei, J. A. 1995, ApJ, 446, 842 [NASA ADS] [CrossRef] [Google Scholar]
 Mihalas, D. 1978, Stellar atmospheres (San Francisco: W. H. Freeman and Co.) [Google Scholar]
 Mitsuda, K., Inoue, H., Koyama, K., et al. 1984, PASJ, 36, 741 [NASA ADS] [Google Scholar]
 Mukai, K., Kinkhabwala, A., Peterson, J. R., Kahn, S. M., & Paerels, F. 2003, ApJ, 586, L77 [NASA ADS] [CrossRef] [Google Scholar]
 Nagel, T., Dreizler, S., Rauch, T., & Werner, K. 2004, A&A, 428, 109 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Narayan, R., Loeb, A., & Kumar, P. 1994, ApJ, 431, 359 [NASA ADS] [CrossRef] [Google Scholar]
 Nauenberg, M. 1972, ApJ, 175, 417 [NASA ADS] [CrossRef] [Google Scholar]
 Nelan, E. P., & Bond, H. E. 2013, ApJ, 773, L26 [NASA ADS] [CrossRef] [Google Scholar]
 Nugis, T., & Lamers, H. J. G. L. M. 2002, A&A, 389, 162 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Olson, G. L., & Kunasz, P. B. 1987, J. Quant. Spectr. Rad. Transf., 38, 325 [NASA ADS] [CrossRef] [Google Scholar]
 Patterson, J. 1981, ApJS, 45, 517 [NASA ADS] [CrossRef] [Google Scholar]
 Patterson, J., & Raymond, J. C. 1985a, ApJ, 292, 550 [NASA ADS] [CrossRef] [Google Scholar]
 Patterson, J., & Raymond, J. C. 1985b, ApJ, 292, 535 [NASA ADS] [CrossRef] [Google Scholar]
 Piro, A. L., & Bildsten, L. 2004a, ApJ, 616, L155 [NASA ADS] [CrossRef] [Google Scholar]
 Piro, A. L., & Bildsten, L. 2004b, ApJ, 610, 977 [NASA ADS] [CrossRef] [Google Scholar]
 Polidan, R. S., & Holberg, J. B. 1984, Nature, 309, 528 [NASA ADS] [CrossRef] [Google Scholar]
 Popham, R. 1997, ApJ, 478, 734 [NASA ADS] [CrossRef] [Google Scholar]
 Popham, R. 1999, MNRAS, 308, 979 [NASA ADS] [CrossRef] [Google Scholar]
 Popham, R., & Narayan, R. 1995, ApJ, 442, 337 [NASA ADS] [CrossRef] [Google Scholar]
 Pringle, J. E. 1977, MNRAS, 178, 195 [NASA ADS] [CrossRef] [Google Scholar]
 Pringle, J. E., & Savonije, G. J. 1979, MNRAS, 187, 777 [NASA ADS] [CrossRef] [Google Scholar]
 Prinja, R. K., & Rosen, R. 1995, MNRAS, 273, 461 [NASA ADS] [CrossRef] [Google Scholar]
 Proga, D., Stone, J. M., & Drew, J. E. 1998, MNRAS, 295, 595 [NASA ADS] [CrossRef] [Google Scholar]
 Rauch, T., Orio, M., GonzalesRiestra, R., et al. 2010, ApJ, 717, 363 [NASA ADS] [CrossRef] [Google Scholar]
 Regev, O. 1983, A&A, 126, 146 [NASA ADS] [Google Scholar]
 Regev, O., & Bertout, C. 1995, MNRAS, 272, 71 [NASA ADS] [CrossRef] [Google Scholar]
 Revnivtsev, M. G., & Gilfanov, M. R. 2006, A&A, 453, 253 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Revnivtsev, M. G., Burenin, R. A., Tkachenko, A. Y., et al. 2012, Astron. Lett., 38, 238 [NASA ADS] [CrossRef] [Google Scholar]
 Revnivtsev, M. G., Suleimanov, V. F., & Poutanen, J. 2013, MNRAS, 434, 2355 [NASA ADS] [CrossRef] [Google Scholar]
 Robertson, J. A., & Frank, J. 1986, MNRAS, 221, 279 [NASA ADS] [CrossRef] [Google Scholar]
 Robinson, E. L., & Nather, R. E. 1979, ApJS, 39, 461 [NASA ADS] [CrossRef] [Google Scholar]
 Shakura, N. I., & Sunyaev, R. A. 1973, A&A, 24, 337 [NASA ADS] [Google Scholar]
 Shaviv, G., & Wehrse, R. 1991, A&A, 251, 117 [NASA ADS] [Google Scholar]
 Sion, E. M., Long, K. S., Szkody, P., & Huang, M. 1994, ApJ, 430, L53 [NASA ADS] [CrossRef] [Google Scholar]
 Sion, E. M., Godon, P., Myzcka, J., & Blair, W. P. 2010, ApJ, 716, L157 [NASA ADS] [CrossRef] [Google Scholar]
 Suleimanov, V., & Poutanen, J. 2006, MNRAS, 369, 2036 [NASA ADS] [CrossRef] [Google Scholar]
 Suleimanov, V., & Werner, K. 2007, A&A, 466, 661 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Suleimanov, V. F., Lipunova, G. V., & Shakura, N. I. 2007, Astron. Rep., 51, 549 [NASA ADS] [CrossRef] [Google Scholar]
 Suleimanov, V. F., Mauche, C. W., Zhuchkov, R. Y., & Werner, K. 2013, in 18th European White Dwarf Workshop., eds. J. Krzesiński, G. Stachowski, P. Moskalik, & K. Bajan, ASP Conf. Ser., 469, 349 [Google Scholar]
 Suleymanov, V. F. 1992, Sov. Astron. Lett., 18, 104 [NASA ADS] [Google Scholar]
 Swank, J. H. 1979, in White Dwarfs and Variable Degenerate Stars, eds. H. M. van Horn, & V. Weidemann, IAU Colloq., 53, 135 [Google Scholar]
 Tylenda, R. 1977, Acta Astron., 27, 235 [NASA ADS] [Google Scholar]
 Tylenda, R. 1981, Acta Astron., 31, 267 [NASA ADS] [Google Scholar]
 van der Klis, M. 1989, ARA&A, 27, 517 [NASA ADS] [CrossRef] [Google Scholar]
 van der Klis, M. 2000, ARA&A, 38, 717 [NASA ADS] [CrossRef] [Google Scholar]
 van Teeseling, A., & Verbunt, F. 1994, A&A, 292, 519 [NASA ADS] [Google Scholar]
 van Teeseling, A., Beuermann, K., & Verbunt, F. 1996, A&A, 315, 467 [NASA ADS] [Google Scholar]
 Velikhov, E. 1959, Sov. Phys. JETP, 36, 1398 [Google Scholar]
 Verner, D. A., & Yakovlev, D. G. 1995, A&AS, 109, 125 [NASA ADS] [Google Scholar]
 Verner, D. A., Ferland, G. J., Korista, K. T., & Yakovlev, D. G. 1996, ApJ, 465, 487 [NASA ADS] [CrossRef] [Google Scholar]
 Wade, R. A., & Hubeny, I. 1998, ApJ, 509, 350 [NASA ADS] [CrossRef] [Google Scholar]
 Warner, B. 1986, Ap&SS, 118, 271 [NASA ADS] [CrossRef] [Google Scholar]
 Warner, B. 2003, Cataclysmic Variable Stars (Cambridge: Cambridge University Press) [Google Scholar]
 Warner, B., & Pretorius, M. L. 2008, MNRAS, 383, 1469 [NASA ADS] [CrossRef] [Google Scholar]
 Warner, B., & Woudt, P. A. 2002, MNRAS, 335, 84 [NASA ADS] [CrossRef] [Google Scholar]
 Zhang, E.H., & Robinson, E. L. 1987, ApJ, 321, 813 [NASA ADS] [CrossRef] [Google Scholar]
All Tables
Parameters of the rings for the models with parameters M_{WD} = M_{⊙} and ω_{WD} = 0.8, 0.6, 0.4, 0.2 ω_{K}.
Parameters of the rings for the models with parameters ω_{WD} = 0.8 ω_{K} and M_{WD} = 1.2 and 0.8 M_{⊙}.
Parameters of the rings for the model with parameters M_{WD} = 0.8 M_{⊙} and ω_{WD} = 0.6 ω_{K}.
Parameters of the strongest emission lines, preliminarily identified in the subtracted spectrum.
All Figures
Fig. 1 Distribution of BL quantities along radius. The data are from the model with M_{WD} = M_{⊙} and ω_{WD} = 0.8 ω_{K}. The position of the WD surface at R_{WD} = 5.48 × 10^{8} cm is shown by the vertical dotted line. Top panel: shown is the effective temperature (solid curve) together with the relative angular velocity (dashed curve), as obtained from the 1D hydrodynamical model. These are the same for the atmosphere models. Middle and bottom panels: Rosseland optical halfthickness, τ_{0}, and geometrical halfthickness, H_{0}, for the modelatmosphere models (solid lines). The distributions obtained from the 1D hydrodynamical model (dotted curves) and the gray model (dashed curves) are also shown. 

Open with DEXTER  
In the text 
Fig. 2 Top panel: local spectra of the hottest ring of the same BL model as in Fig. 1 computed with (solid curve) and without (dashed curve) spectral lines taken into account. The blackbody spectrum corresponding to the effective temperature is shown by the dotted curve. Bottom panel: temperature structures of the model with lines (solid curve) and the gray model (dashed curve). The central temperature of the hydrodynamical 1D model is shown by the dotted line. 

Open with DEXTER  
In the text 
Fig. 3 Top panel: sound speed (dashed curve) and wind velocity (solid curve) distributions along depth in the ring model shown in Fig. 2. Middle panel: distributions of gravity (dashed curve) and radiative acceleration (solid curve) along depth. The position of the sonic point is marked by the dotted vertical line. Bottom panel: depths where the emergent spectrum forms (τ_{λ} = 1). The position of the sonic point is marked by the dashed horizontal line. 

Open with DEXTER  
In the text 
Fig. 4 Spectra of the BL model M = M_{⊙}, ω_{WD} = 0.8 ω_{K} for three inclination angles relative to the line of sight. The corresponding blackbody approximations are shown by dashed curves. 

Open with DEXTER  
In the text 
Fig. 5 Detail of the integral emergent fluxes of the same BL model as in Fig. 4 computed for i = 0° (rotation does not affect the spectrum, thin black curve), and 15° (thick red curve). 

Open with DEXTER  
In the text 
Fig. 6 Comparison of the spectra of the same BL model (see Fig. 4) computed using five (dotted curve) and twenty (solid curve) rings. 

Open with DEXTER  
In the text 
Fig. 7 Comparison of the BL model spectrum from Fig. 4 (solid curve) with the spectrum of the hottest ring (dotted curve) normalized to the flux at 100 Å. 

Open with DEXTER  
In the text 
Fig. 8 Effective temperature distributions (top panel, the temperatures of the considered rings are marked by dots) and model spectra (bottom panel) for the models with different angular velocities (0.2, 0.4, 0.6 and 0.8 ω_{K}) and fixed WD mass (M_{WD} = M_{⊙}). 

Open with DEXTER  
In the text 
Fig. 9 Spectra of the same models as in Fig. 8 in the observed wavelength range. The spectra in the top panel are normalized to the ω_{WD} = 0.6 ω_{K} model spectrum at 100 Å. 

Open with DEXTER  
In the text 
Fig. 10 Comparison of BL model spectra for three WD masses (0.8, 1, 1.2 M_{⊙}) and fixed relative angular velocity (ω_{WD} = 0.8 ω_{K}). The spectra in the top panel are normalized to the M_{WD} = M_{⊙} model flux at 100 Å. 

Open with DEXTER  
In the text 
Fig. 11 Top panel: comparison of the observed SS Cyg soft Xray spectrum (solid curve) with the BL model spectrum for a 1 M_{⊙} WD and relative angular velocity ω_{WD} = 0.8 ω_{K} (dashed curve). Bottom panel: subtracted (observed minus BL model) spectrum (solid curve) together with the homogeneous slab model spectrum (dashed curve). The emission continuum of Ne VII below 60 Å and the strongest identified emission lines are marked. 

Open with DEXTER  
In the text 
Fig. 12 Scheme of the BL outflow. 

Open with DEXTER  
In the text 
Fig. 13 Comparison of the BL model spectra for a 0.8 M_{⊙} WD and relative angular velocity ω_{WD} = 0.6 ω_{K} (dashed curve) and for a 1 M_{⊙} WD and relative angular velocity ω_{WD} = 0.8 ω_{K} (solid curve). The flux of the latter spectrum is multiplied by 2.5. 

Open with DEXTER  
In the text 
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