Modifying the Standard Disk Model for the Ultraviolet Spectral Analysis of Disk-dominated Cataclysmic Variables. I. The Novalikes MV Lyrae, BZ Camelopardalis, and V592 Cassiopeiae

MV Lyr has an inclination i ∼ 10° (Schneider et al. 1981; Skillman et al. 1995; Linnell et al. 2005) and zero reddening (Bruch & Engel 1994). It is a VY Scl novalike system, i.e., spending most of its time in a high state and occasionally undergoing short-duration drops in brightness, or "low state," when accretion is almost shut off and the WD is revealed in the UV. It has a distance d ∼ 500 pc and a WD mass of M_wd ∼ 0.73–0.80 M_⊙ (Hoard et al. 2004; Linnell et al. 2005; Godon et al. 2012). Compared to other CVs, the values of these parameters are relatively well known. The orbital period of the binary is P = 3.19 hr and the mass ratio was found to be $q={M}_{2\mathrm{nd}}/{M}_{\mathrm{wd}}=0.4$ (Schneider et al. 1981; Skillman et al. 1995). The system parameters are listed in Table 1.

MV Lyr spends most of its time in the high state, at a visual magnitude of ∼12–13, when the emission is primarily from the accretion disk. From time to time it drops into a low state, reaching a magnitude of ∼17.5, when the emission is possibly entirely from the WD. MV Lyr also has periods during which it is in an intermediate state at a magnitude of ∼14–15. MV Lyr was observed with FUSE and IUE in a low state, with IUE in an intermediate state, and with FUSE and HST/STIS in a high state. The IUE observations of MV Lyr during its low state revealed a hot WD possibly reaching 50,000 K (Szkody & Downes 1982) or higher (Chiappetti et al. 1982). These results were later confirmed with a FUSE spectroscopic analysis in a low state (Hoard et al. 2004), during which the mass accretion rate was estimated to be no more than $\dot{M}\approx 3\times {10}^{-13}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$ and the WD temperature was found to be 47,000 K. A follow-up analysis to study the different states of MV Lyr was carried out by Linnell et al. (2005), who found that standard disk models did not fit the observed spectra. Their models improved with the truncation of the inner disk, an isothermal radial temperature profile in the outer disk, or both, and gave a mass accretion rate of the order of 3 × 10⁻⁹ M_⊙ yr⁻¹ for the high state and 1 × 10⁻⁹ M_⊙ yr⁻¹ for the intermediate state.

The departure from the standard disk model is more pronounced in the intermediate state and most evident in the longer wavelengths of IUE. Since the high state was only observed down to ∼1800 Å, in our analysis we only model the system in the low and intermediate states extending above 3000 Å. The modeling of the low state is first carried out (see Section 5.1) to demonstrate the numerical method and to re-derive the system parameters. The observation log is recapitulated in Table 2. The spectra are shown in Figure 1 together with a standard disk model for comparison. The shallow slope of the IUE spectra in the intermediate state appears clearly.

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All of the FUSE spectra in this work were calibrated with the latest and final version of CalFUSE (Dixon et al. 2007), and processed using our suite of FORTRAN programs, UNIX scripts, and IRAF procedures written for this purpose. The details can be found in Godon et al. (2012).

BZ Cam is a VY Scl NL type (Garnavitch & Szkody 1988; Greiner et al. 2001), with a relatively small reddening E(B–V) = 0.05 (Prinja et al. 2000), an orbital period of 221 minutes (Patterson et al. 1996), a distance of 830 ± 160 pc (Ringwald & Naylor 1998), and a moderately low inclination ∼12° < 40° (Ringwald & Naylor 1998). Its WD mass is unknown, but in our analysis we assume M_wd ≈ 0.4 M_⊙ ≤ M_wd ≤ 0.7 M_⊙ (Greiner et al. 2001). BZ Cam is not a typical CV in that it is surrounded by a faint emission bow-shock nebula (Ellis et al. 1984; Krautter et al. 1987; Greiner et al. 2001), which is proof of an outflow (Patterson et al. 1996). All the system parameters of BZ Cam are listed in Table 1 with their references.

BZ Cam spends most of its time in a high state at a magnitude of ∼12–13. It has been shown to drop occasionally to a magnitude of ∼14. All the UV spectra of BZ Cam were collected in the high state, but show some small variations in the continuum flux level. The first ultraviolet (UV) spectra of BZ Cam were obtained with IUE (Krautter et al. 1987; Woods et al. 1990, 1992; Griffith et al. 1995). We retrieved here 13 IUE spectra from a single epoch (1988 November 19 and 1988 November 20) with the same continuum flux level. They consist of six IUE LWP spectra and seven IUE SWP spectra, which we co-added and then combined (see Table 2). The IUE (SWP and LWP) and optical continuum is consistent with a ∼12,500 K Kurucz LTE model atmosphere with $\mathrm{log}g=2.5$ (Prinja et al. 2000), but it cannot be attributed to the WD, because of its small emitting surface area. The IUE spectra of BZ Cam has a relatively shallow slope when compared to an accretion disk. BZ Cam also has a FUSE spectrum (Froning et al. 2012), which was never previously modeled with realistic disk or WD atmosphere model spectra. The FUSE spectrum is affected by interstellar (ISM) absorption, mainly molecular hydrogen. The FUSE and IUE spectra of BZ Cam are shown (using a coarse binning for clarity) in Figure 2 together with a standard disk model and a 12,500 K stellar atmosphere (with $\mathrm{log}g=8$) spectrum for comparison. The continuum flux level in the FUSE spectrum is slightly lower than the continuum flux level of the combined IUE spectrum in the region where the two spectra overlap. This difference is likely due to a small variation in the flux of BZ Cam, although we cannot exclude the possibility that it could also be due, at least in part, to the fact that the data were obtained with different telescopes, gratings, detectors, and software.

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V592 Cas is a UX UMa novalike subtype and has never been observed in a low brightness state. It has an inclination of i = 28° and a mass ratio of 0.19 (Huber et al. 1998), and its orbital period is P_orb = 2.76 hr (Taylor et al. 1998; Witherick et al. 2003). It shows evidence for a bipolar wind outflow (Witherick et al. 2003; Prinja et al. 2004; Kafka et al. 2009) with velocities reaching up to v_wind ∼ 5000 km s⁻¹ (Kafka et al. 2009). The reddening toward the system is E(B–V) = 0.22 (Cardelli et al. 1989), and its distance is about 240–360 pc (Taylor et al. 1998). The mass accretion rate of the system is believed to be of the order of ≈1 × 10⁻⁸ M_⊙ yr⁻¹ (Taylor et al. 1998; Hoard et al. 2009). In the present work, we assume the WD to have a standard CV WD mass of ∼0.8 M_⊙ and a temperature of 45,000 K (Hoard et al. 2009).

V592 Cas was observed with IUE on 1981 December 5. We retrieved the two LWR spectra and two of the four SWP spectra that were obtained, and combined and co-added them to generate a spectrum with a spectral range from ∼1150 Å to ∼3200 Å. With a flux of a few 10⁻¹³ erg s⁻¹ cm⁻² Å⁻¹, the spectrum has a relatively good signal in the SWP segment but it is rather noisy in the LWR segment, as shown in Figure 3 (in red). V592 Cas has three FUSE data sets, obtained on 2003 August 5, 7, and 8. The data sets all have the same quality and about the same exposure time and reveal the same spectrum. We decided to retrieve the first data set (D1140101) and extracted a final spectrum following the same procedure we used for BZ Cam and MV Lyr. The FUSE spectrum of V592 Cas has the same continuum flux level as the IUE spectrum in the region where the two spectra overlap. Like the FUSE spectrum of BZ Cam, the FUSE spectrum of V592 Cas is heavily affected by ISM molecular hydrogen lines, appearing in Figure 3 (in black) as if the spectrum had literally been sliced.

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In Figure 3 we also plot two standard disk models for comparison. The FUSE continuum is in better agreement with the 10⁻⁸ M_⊙ yr⁻¹ model, while the IUE continuum slope is even shallower than the 10⁻⁹ M_⊙ yr⁻¹ model. These indicate that the combined FUSE + IUE spectrum does not agree very well with the standard disk model. However, the discrepancy is not as strong as for MV Lyr and BZ Cam.

In the standard disk theory (Pringle 1981), the total accretion luminosity that can be released by accretion onto a star is given by

$\begin{eqnarray}&&{L}_{\mathrm{acc}}=\displaystyle \frac{{{GM}}_{\mathrm{wd}}\dot{M}}{{R}_{\mathrm{wd}}},\end{eqnarray} \tag{ 1 }$

where G is the gravitational constant, $\dot{M}$ the mass accretion rate, M_wd the mass of the accreting star (here a WD), and R_wd its radius.

In disk systems, accretion energy is released through the disk. All current disk models are based on the Shakura−Sunyaev "alpha" disk model known as the standard disk model (Shakura & Sunyaev 1973; Lynden-Bell & Pringle 1974). The disk is assumed to be geometrically thin (in the vertical dimension), and the energy dissipated between adjacent rings of matter (Φ_ν, the dissipation function due to the shear) is radiated locally in the vertical direction, $\sigma {T}_{\mathrm{eff}}^{4}={{\rm{\Phi }}}_{\nu }$. The viscosity ν is due to turbulence and has been shown to come from a magneto-rotational instability (Balbus & Hawley 1991). The turbulent viscosity is parametrized with an a priori unknown parameter α (α < 1), which is the reason the standard disk model is also called the alpha disk model. The standard disk model is obtained from the steady-state Navier–Stokes equations in cylindrical coordinates (R, ϕ, z)). The equations reduce to one dimension (R) after integrating in the vertical dimension (z) and assuming axisymmetry (∂ϕ/∂ = 0). Using the conservations of mass (for $\dot{M}$) and angular momentum and imposing the boundary condition that the angular velocity gradient vanishes (∂Ω/∂R = 0) at the inner boundary R = R_wd, one obtains the standard disk model (Pringle 1981). In that approximation, the BL between the star and disk is geometrically thin, and its thickness can be neglected δ_BL ≪ R_wd.

We modify the standard disk model by truncating the inner disk at a radius R₀ > R_wd. In order to do this correctly, one cannot just remove the inner disk between R_wd and R₀ from a standard disk model starting at R_wd. Instead, one has to generate a standard disk model starting at R = R₀. We give below several reasons for having a disk start at R₀ rather than R_wd.

The optically thin BL simulations of Narayan & Popham (1993) and Popham (1999) show that the optically thin BL has the size δ_BL ≈ R_wd and that the angular velocity gradient vanishes at the outer edge of the BL. Namely, the no-shear boundary condition ∂Ω/∂R = 0 has to be imposed at R = R₀ = R_wd + δ_BL > R_wd.

If the inner hole in the disk is due to a weakly magnetized WD truncating the disk (Hameury & Lasota 2002) at R₀ (as in IPs), one still has the same new boundary conditions, as R₀ is now the corotation radius, which by definition is the radius where ∂Ω/∂R = 0, since Ω = constant there.

If the inner hole in the disk is due to evaporation (Hameury et al. 2000), the same new boundary conditions can also be imposed as the disk only starts at the outer radius (now R₀) of the evaporated inner disk.

Also, it is important to note that the radius of the WD is expected to increase as the WD temperature increases. For example, a 10,000 K WD of mass 0.7 M_⊙ has a radius R_wd = 7.9 × 10⁸ cm, and at 50,000 K, the radius increases by 16% to R_wd = 9.2 × 10⁸ cm (Wood 1995). The increase in radius is even more pronounced for a smaller mass WD and basically reaches a factor of 2 for a 0.4 M_⊙ WD, with a zero-temperature radius of R_wd = 1.1 × 10⁹ cm versus a 50,000 K radius of R_wd = 2 × 10⁹ cm (Wood 1995). Since the VY Scl systems are notorious for having a temperature much hotter than other CVs at the same period, the change in radius cannot be ignored. So far, most standard disk model spectra were computed assuming a constant (zero-temperature) radius (Wade & Hubeny 1998; Godon et al. 2012), thereby not taking into account the increased WD radius and possibly overestimating the disk temperature. Modeling the disk properly requires that the WD temperature is known, and requires a fine-tuning of the modeling such as the one performed in Linnell et al. (2008).

The modified disk model can therefore be applied to different inner disk structures, including the magnetically truncated disks in IP systems and around hot CV WDs to provide a more realistic approach for the modeling of the disk.

Our modified disk model differs from the standard disk model in that we now impose the no-shear boundary condition at r = R₀ > R_wd. Namely,

$\begin{eqnarray}&&\displaystyle \frac{\partial {\rm{\Omega }}}{\partial R}=0,\,\,\,R={R}_{0},\end{eqnarray} \tag{ 2 }$

and, following Pringle (1981), we assume a Keplerian velocity Ω = Ω_K for R ≥ R₀.

With these assumptions and boundary conditions, one obtains the disk radial temperature profile

$\begin{eqnarray}&&{T}_{\mathrm{eff}}(R)={\left\{\displaystyle \frac{3{{GM}}_{\mathrm{wd}}\dot{M}}{8\pi \sigma {R}^{3}}\left[1-\sqrt{\displaystyle \frac{{R}_{0}}{R}}\right]\right\}}^{1/4},\end{eqnarray} \tag{ 3 }$

where the angular velocity in the disk has been substituted with its Keplerian value. For convenience, this expression can be written as

$\begin{eqnarray}{T}_{\mathrm{eff}}(x) & = & {T}_{0}{x}^{-3/4}{(1-{x}^{-1/2})}^{1/4},\,\mathrm{with}\,{T}_{0}=\mathrm{64,800}\,{\rm{K}}\\ & & \times {\left[\left(\displaystyle \frac{{M}_{\mathrm{wd}}}{1{M}_{\odot }}\right)\left(\displaystyle \frac{\dot{M}}{{10}^{-9}{M}_{\odot }{\mathrm{yr}}^{-1}}\right){\left(\displaystyle \frac{{R}_{0}}{{10}^{9}\mathrm{cm}}\right)}^{-3}\right]}^{1/4},\end{eqnarray} \tag{ 4 }$

where x = R/R₀. This model gives a maximum temperature (${T}_{\max }=0.488{T}_{0}$) at x = 1.36, and T = 0 K at R = R₀ (x = 1).

The expression for the temperature of the modified disk model is the same as that for the standard disk model, except for the star radius R_wd, which has simply been replaced by the inner disk radius R₀. For example, for an inner disk radius twice as large as the star radius (R₀ = 2 R_wd), the maximum effective surface temperature in the disk (reached at R = 1.36 R₀) has dropped to ∼65% of the value it has in the standard disk model (at R = 1.36 R_wd). In Figure 4 we show the temperature profile for a modified disk model for different values of the truncated radius R₀. The mass of the WD is 0.8 M_⊙, and the mass accretion rate is 1 × 10⁻⁹ M_⊙ yr⁻¹. The radius R₀ is given in units of the WD radius R_wd (which is itself obtained from the WD mass–radius relation; e.g., Wood 1995). From Figure 4, it is clearly apparent that truncating a standard disk model by solely removing the region R < 2 R_wd gives a disk model with a much higher temperature in the inner disk than generating a model starting at R₀ = 2 R_wd. As the size of the BL is possibly not negligible and as the WD can easily increase its radius with increasing temperature, the standard disk model can overestimate the temperature in the inner disk by a large fraction.

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Although our modified disk model is actually a standard disk model in which the radius of the star R_wd has been replaced by the inner radius of the disk R₀ (where R₀ > R_wd), we use in this paper the terms modified disk and standard disk to differentiate between the two.

The disk luminosity L_disk is given by integrating Φ_ν (or $\sigma {T}_{\mathrm{eff}}^{4}$) over the surface of the disk

$\begin{eqnarray}&&{L}_{\mathrm{disk}}=2{\int }_{{R}_{0}}^{\infty }{{\rm{\Phi }}}_{\nu }2\pi {RdR}=\displaystyle \frac{1}{2}\displaystyle \frac{{{GM}}_{\mathrm{wd}}\dot{M}}{{R}_{0}}.\end{eqnarray} \tag{ 5 }$

In the standard disk model, the disk luminosity amounts to half the accretion energy; in the modified disk model, it is smaller than half the accretion energy. For example, for R₀ = 2 R_wd, the disk luminosity is only half the disk luminosity expected for a standard non-truncated disk. The important point is that a truncated disk cannot be generated by simply truncating a standard disk model, but it has to be obtained as a standard disk model in which R_wd is replaced by R₀, which produces a much lower disk temperature and luminosity. The idea of replacing R_wd by R₀ was discussed briefly in the modeling of IX Vel (Long et al. 1994) but was never followed through. Similarly, the modeling of MV Lyr by Linnell et al. (2005) includes a truncated disk, but with the "usual" temperature profile of the standard disk model (R_wd was not replaced by R₀; instead, the standard disk was just truncated).

The remaining fraction of the accretion energy is dissipated at the inner edge of the disk, where the disk meets the surface of the star (r = R_wd), in the so-called BL. The fast (Keplerian) rotating (Ω_K(R₀)) matter adjusts itself to the slowly rotating stellar surface (Ω_wd) and dissipates its remaining kinetic energy. The luminosity of the BL, L_BL, in the standard disk model, is of the same order of magnitude as L_acc/2, although it is usually smaller as some of the energy goes into spinning up the WD (Kluźniak 1987). This is true for optically thick BLs.

X-ray observations of disk-dominated CVs (e.g., van Teeseling et al. 1996; Balman et al. 2014) show that the BL is optically thin and that the X-ray luminosity is much smaller than the disk luminosity; the BL is underluminous. For the optically thin BL, one can only refer to the work of Narayan & Popham (1993) and Popham (1999), and to observations.

If the disk emits less than half the accretion energy, and the BL luminosity (in the X-ray) is itself only a fraction of the disk luminosity, one might ask, where does the remaining energy go? When an accretion flow cannot cool efficiently, due to being optically thin, the energy is advected with the flow into the outer layer of the star (Abramowicz et al. 1995). The advected BL energy will then increase the WD temperature and will likely drive an outflow as in ADAFs (Narayan & Yi 1995). This is supported by three observational facts. (i) The WD temperature of NLs is rather elevated when compared to other CVs with the same period. (ii) Many NL systems exhibit hot, fast wind outflows seen in the P Cygni line structure and blueshifted absorption. These winds originate in the inner disk/BL region but their source of energy has not yet been identified. Some CV systems even have visible ejected nebular material (e.g., BZ Cam, Ellis et al. 1984). (iii) Recent X-ray observations of NLs (Balman et al. 2014) suggest that high-state NLs may have optically thin BLs merged with ADAF-like flows and/or hot coronae. Balman et al. (2014) showed that the observed discrepancy between the X-ray/BL and disk luminosity can be accounted for if the BL energy is heating the WD through advection (T_wd ∼ 50,000 K) and is also used to drive the wind outflows. The efficiency of the BL in radiating its energy is then greatly reduced (<0.01), consistent with the observed ratio of the X-ray to the disk UV-optical luminosities.

For this reason, we expect NLs to be modeled with the modified disk model and the addition of a hot WD, itself possibly with an inflated radius.

In CVs, the accreting WD is the more massive component and the Roche lobe radius is always larger than half the binary separation, i.e., >a/2. Due to the impact of the stream (from the first Lagrange point L1) on the rim of the disk and due to tidal interaction, the outer disk is expected to have a temperature larger than dictated by both the standard and modified disk models (Buat-Ménard et al. 2001). We assume here that the outer region of the disk has a temperature of 10,000 K to 12,000 K. We note that such a practice is not uncommon, and some authors (e.g., Linnell et al. 2005) have considered disk models with an elevated temperature (∼10–12,000 K) in the outer region of the disk.

We use the FORTRAN suite of codes TLUSTY, SYNSPEC, ROTIN, and DISKSYN (Hubeny 1988; Hubeny et al. 1994; Hubeny & Lanz 1995) to generate synthetic spectra of stellar atmospheres and disks. The codes include the treatment of hydrogen quasi-molecular satellite lines (low temperature) and LTE and NLTE options (high temperature). SYNSPEC generates a continuum with absorption lines. In the present case, we do not generate emission lines. For disk spectra, we use solar abundances, and for stellar spectra, we vary the abundances as needed. An introductory guide, a reference manual, and an operational manual for TLUSTY and SYNSPEC have just been released and are available with the full details of the codes (Hubeny & Lanz 2017a, 2017b, 2017c).

The code TLUSTY is first run to generate a one-dimensional (vertical) stellar atmosphere structure for a given surface gravity, effective temperature, and surface composition of the star. Computing a single model is an iterative process that needs to converge. In the present case, we treat hydrogen and helium explicitly, and treat nitrogen, carbon, and oxygen implicitly (Hubeny & Lanz 1995).

The code SYNSPEC is then run, using the output stellar atmosphere model from TLUSTY as an input, and it generates a synthetic stellar spectrum over a given (input) wavelength range (it has the capabilities to cover a spectral range from below 900 Å into the optical). The code SYNSPEC derives the detailed radiation and flux distribution of the continuum and lines, and generates the output spectrum (Hubeny & Lanz 1995). SYNSPEC has its own chemical abundances input to generate lines for the chosen species. For temperatures above 35,000 K, the approximate NLTE treatment of lines is turned on in SYNSPEC.

Rotational and instrumental broadening as well as limb darkening are then reproduced using the routine ROTIN. In this manner, we have already generated a grid of WD synthetic spectra covering a wide range of temperatures and gravities with solar composition (see http://synspecat.weebly.com/).

The disk spectra are generated by dividing the disk into rings, each with a given radius r_i and temperature (T(r_i)), and the density and effective vertical gravity obtained from the modified disk model for a given stellar mass, inner and outer disk radii, and mass accretion rate.

The code TLUSTY generates a one-dimensional vertical structure for each disk ring. The input parameters are the mass accretion rate, the mass of the accreting star, the radius of the star, and the radius of the ring. The radius of the star in our modified disk model is set to be the inner radius of the disk R₀.

SYNSPEC is then used to create a spectrum for each ring, and the resulting ring spectra are integrated into a disk spectrum using the code DISKSYN, which includes the effects of (Keplerian) rotational broadening, inclination, and limb darkening.

For disk rings below 10,000 K, we use Kurucz stellar spectra of appropriate temperature and surface gravity.

In Figure 5, we present the synthetic spectra of two disks generated with the suite of codes: a standard disk model (in black) and a modified disk model (in red). Both models have a 0.55 M_⊙ WD accreting at a rate of $\dot{M}={10}^{-9}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$, and are displayed for an inclination i = 40°. The standard disk has an inner radius at R₀ = R_wd, while the modified disk has an inner radius at R₀ = 2 R_wd. The slope of the continuum of the modified disk is almost flat.

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The Low State. The modeling of the FUSE spectrum of MV Lyr in the low state has been presented elsewhere (Godon et al. 2012) and gives identical results to the modeling of the combined FUSE + IUE spectrum presented here, which covers a much larger wavelength range. However, it is important to re-derive and confirm the distance to MV Lyr, its WD mass, radius, temperature, and rotational velocity, because these are important parameters serving as input for the modeling of the intermediate state. Namely, these parameters are used as constrains in the modeling of the combined FUSE + IUE spectrum of MV Lyr in the intermediate state.

While the FUSE spectrum presents many absorption lines commonly detected in the atmosphere of accreting WDs (see Godon et al. 2012 for details), the IUE (SWP+LWP) spectrum is noisier and presents only a continuum, in which the only identified absorption feature is the C iv 1550 line. The FUSE and IUE spectra match in their flux level and slope, in spite of the fact that they were obtained with different telescopes more than 20 years apart. This is possibly a sign that MV Lyr had reached a similar (if not identical) low state in which the WD completely dominates the FUV (Hoard et al. 2004).

Our fitting results yield (Godon et al. 2012) a WD temperature T_wd = 44–47,000 K, with a projected stellar rotational velocity ${V}_{\mathrm{rot}}\sin i=150\mbox{--}250\,\mathrm{km}\,{{\rm{s}}}^{-1}$, chemical abundances Z = 0.1–1.0 Z_⊙, and stellar surface gravity $\mathrm{log}g=8.06\mbox{--}8.28$ (M_wd = 0.7–0.8 M_⊙ with R_wd = 9010–7450 km). The resulting distance we obtained is ∼464 pc for a 0.8 M_⊙ WD and ∼560 pc for a 0.7 M_⊙ WD. In Figure 6 we present a 45,000 K WD fit as one of the solutions we obtained. The remarkable characteristic of this model fit is that, although it was originally generated to fit the FUSE spectrum alone (Godon et al. 2012), as shown in Figure 6, it actually also fits the IUE spectrum all way to the longest wavelengths, ∼3000 Å (see Figure 7; the region beyond 3000 Å is very noisy and has been discarded). From this fit, one obtains relatively accurate values for some of the important system parameters.

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These values of the parameters are consistent with the analysis of Hoard et al. (2004), who obtained T_wd = 47,000 K with $\mathrm{log}g=8.25$ for a non-rotating WD, and T_wd = 44,000 K with $\mathrm{log}g=8.22$ for a WD rotating at 200 km s⁻¹ with elemental abundances of C = 0.5, N = 0.5, and Si = 0.2 (solar), and a derived distance of 505 ± 50 pc.

The Intermediate State. MV Lyr was caught in an intermediate state with IUE, covering that region of the spectrum where the continuum slope departs strongly from a standard disk model all the way to ∼3200 Å. In order to model the intermediate state spectrum, we use the parameters we derived from fitting the low state: we adopt a WD mass of M_wd = 0.73 M_⊙, with a temperature of ∼45,000 K, a radius R_wd = 8,500 km, and a distance of ∼500 pc. We use the system inclination i = 10° from the literature (see Table 1).

We first attempt to fit the combined IUE SWP+LWP spectrum with a standard disk model, where the only varying parameter is the mass accretion rate (since the inclination and WD mass are fixed). The inner disk radius is simply R_wd and the outer radius of the disk is placed at a distance a/3, where a is the binary separation. Using the system parameters, we have a/3 ∼ 30 R_wd. We also include the expected contribution of the 45,000 K WD. In order to fit the distance to the system (by scaling the theoretical spectrum to the observed one), we find a mass accretion rate of 8 × 10⁻¹⁰ M_⊙ yr⁻¹. However, this model does not fit the observed spectrum at all (see Figure 8)—the theoretical spectrum is too "blue" when compared to the observed spectrum, consistent with the statistical analysis of Puebla et al. (2007).

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Next, we fit the observed spectrum with modified standard disk models. For each given mass accretion rate, we built disks with an increasingly larger inner radius, R₀/R_wd = 1.0, 1.2, 1.5, 1.7, 2.0, 2.5, and 3.0. We kept the outer disk radius at 30 R_wd and set the outer disk at a temperature of 10,000, 11,000, or 12,000 K. The width of this isothermal outer disk region was also varied from 1 R_wd to 10 R_wd (i.e., to a maximum of one-third of the disk radial extent). The increase in the inner disk radius produces a decrease in the flux that is more pronounced in the shorter wavelength region of the spectrum, while the setting of the outer isothermal disk region produces a relative increase in the flux in the longer wavelength region. We find that a best fit is obtained for a mass accretion rate of 2.4 × 10⁻⁹ M_⊙ yr⁻¹ with an inner hole of size 2.0 R_wd and an outer disk, from 20 R_wd to 30 R_wd, set with a temperature of 10,000 K. This model is presented in Figure 9 and also includes the contribution of a 45,000 K WD. The strong emission lines are shown in blue; there are more emission lines on the edges of the Lyα profile which we are not modeling, as they form in an optically thin medium. The fit to the continuum is much improved, when compared to the standard disk model.

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BZ Cam has a FUSE spectrum and an IUE spectrum that do not have exactly the same flux level; in addition, the IUE spectrum is consistent with a rather cold component, while the FUSE spectrum seems to be consistent with a hotter component. Consequently, we first model these two spectra separately, and next we try to model the two spectra in a self-consistent manner.

Modeling the FUSE Spectrum. The FUSE spectrum shows the presence of a hot component contributing flux all the way to the shortest wavelengths of FUSE. We try a standard disk model, assuming M_wd = 0.8 M_⊙, and we find that the mass accretion rate has to be as large as 10⁻⁸ M_⊙ yr⁻¹ to produce the observed flux in the short wavelengths of FUSE. This model, however, gives a distance of more than 2 kpc, i.e., more than twice the accepted distance to the system. As we decrease the WD mass to 0.55 M_⊙, the distance decreases slightly to 1.8 kpc, and for a 0.35 M_⊙ WD, the distance is still too large; however, the flux in the shorter wavelengths of the theoretical spectrum is now too low. For a slightly lower mass accretion rate of 10^−8.5 M_⊙ yr⁻¹, only the larger WD model with M_wd = 0.8 M_⊙ provides enough flux in the short wavelengths, but here, too, the distance is too large and reaches 1.4 kpc.

These results show that the single-disk model does not provide a satisfactory fit. Since our modified disk model provides colder disks, we do not try to use such a modeling for the FUSE spectrum, as it would not provide enough flux in the shorter wavelengths.

Instead, we try single-WD models. The best-fit WD model has M_wd = 0.4 M_⊙, T_wd = 45,000 K, solar composition, and ${V}_{\mathrm{rot}}\sin (i)=200\,\mathrm{km}\,{{\rm{s}}}^{-1}$, and the distance obtained from the fit is 791 pc, when the WD radius is set to R_wd = 18,540 km. This model is shown in Figure 10 and includes a basic modeling of the ISM molecular hydrogen absorption lines (see Godon et al. 2007 for details). In the short wavelengths (∼960 Å upper panel), the model has too much flux, while in the longer wavelengths (lower panel), the model has too little flux. A 40,000 K WD gives a distance of 654 pc while a 50,000 K WD gives a distance of 933 pc. We note that at a temperature of about 50,000 K, the radius of the 0.4 M_⊙ WD model (∼2 × 10⁹ cm) is about twice that of the zero-temperature model (∼1 × 10⁹ cm).

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Next, we increase the WD mass to 0.7 M_⊙, and accordingly decrease the radius to ∼0.9 × 10⁹ cm. We find that the scaled distance from the fit becomes too short for the same temperature, and higher temperature models (T > 50,000 K) do not provide the best fit. The fit of the 45,000 K 0.7 M_⊙ WD is as good as the 0.4 M_⊙ fit with the same temperature, and indicates that a 0.7 M_⊙ WD with a radius of ∼2 × 10⁹ cm provides a valid solution, indicating that the radius of the WD might be inflated.

We also try to add a standard disk model to the WD disk model, but we find that the disk model deteriorates the fit and increases the distance well above 1 kpc.

To summarize, the FUSE spectrum of BZ Cam is best fitted with a single WD with a temperature of 45,000 K ± 5000 K, and a mass of 0.4–0.7 M_⊙ (from the literature) as long as the WD radius is ∼2 × 10⁹ cm (to fit the known distance to the system). Since this radius really corresponds to a 0.4 M_⊙ WD at T ∼ 50,000 K (see Section 3.2), this is the WD mass we adopt as the best solution.

Modeling the IUE Spectrum. The IUE spectrum of BZ Cam was modeled with a 12,500 K Kurucz model with $\mathrm{log}g=2.5$ by Prinja et al. (2000), indicating that a standard disk model does not fit the spectrum. Therefore, we do not attempt a standard disk model but instead we use a modified disk model.

We assume a WD mass of 0.4 M_⊙ and a low inclination (i = 10° and i = 20°), and we vary the mass accretion rate from 10⁻¹⁰ M_⊙ yr⁻¹ to 10⁻⁸ M_⊙ yr⁻¹. We assume a WD radius R_wd = 1.15 × 10⁹ cm, which actually corresponds to a 10,000 K 0.4 M_⊙ WD (see Section 3.2), and we vary the inner disk radius R₀ from 1 R_wd to 3 R_wd. We keep the outer part of the disk at an isothermal temperature of 10,000, 11,000, or 12,000 K.

We find that the best modified disk model fit has a mass accretion rate $\dot{M}={10}^{-8.5}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$, i = 20°, and a disk inner radius R₀ = 2 R_wd (=23,000 km), and the outer half of the disk is set to an isothermal temperature T = 10,000 K. This outer half of the disk has a surface area three times larger than the inner half of the disk, making this modified disk model resemble an isothermal disk model. However, in order for this modified disk model to agree with the known distance (830 ± 160 pc), the outer disk radius had to be set at ∼0.5 ± 0.1a, corresponding to R_d ≈ 22 R_wd. Although such a disk radius is about twice the accepted radius (0.3a) in a CV binary, it is still within the limits of the Roche lobe. Choosing an outer disk radius smaller than this value decreases the scaled distances. The fit is shown in Figure 11.

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Since this modified disk model is close to an isothermal disk, we next try an isothermal disk model, with the same inner and outer radii. The isothermal disk model gives very similar results when its temperature is set to 11,000 K. In this model, shown in Figure 12, the deficiency in flux in the shorter wavelength range (the Lyα region) is a little more pronounced than in the modified disk model. In the isothermal disk model, the exact value of the inner disk radius, i.e., whether it is 1 or 2 R_wd, has very little effect on the results as it only changes the surface area of the disk by a tiny fraction. Also, the value of the WD mass is not taken into account. However, when integrating the luminosity over the surface of the isothermal disk, the resulting mass accretion rate corresponds to 3.7 × 10⁻⁹ M_⊙ yr⁻¹ for a 0.4 M_⊙ mass WD and 2.7 × 10⁻⁹ M_⊙ yr⁻¹ for a 0.7 M_⊙ mass WD.

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Self-Consistent Modeling of the FUSE and IUE Spectra. Since the FUSE and IUE spectra were obtained years apart, in slightly different brightness states, and with different telescopes/instruments, we do not expect them to match and we do not attempt to combine them. However, when modeling each spectrum separately (IUE or FUSE), we must take into consideration the results obtained from the modeling of the other spectrum (FUSE or IUE, respectively). Namely, we now model the IUE spectrum with a cold disk and a hot WD model, similar but not identical, to the WD model obtained from the modeling of the FUSE spectrum, and we model the FUSE spectrum with a hot WD and a cold disk model, similar but not identical, to the cold disk model obtained from the modeling of the IUE spectrum.

For the disk model, we chose the isothermal disk model as it is easier to parametrize, and gives results similar to the modified disk model. Namely, the WD mass does not enter the disk model, and the WD radius, taken as the inner radius of the disk, has a minimal effect on the results. For the WD model, we chose a 0.4 M_⊙ WD with a radius increasing with temperature. As we have mentioned earlier, only the WD radius and temperature affect the results.

For the FUSE spectrum, we find that the 45,000 K WD model, with a radius of 18,540 km, is improved significantly with the addition of a cold isothermal disk with T = 11,000 K. The cold disk contributes mainly in the longer wavelengths of the FUSE spectrum and helps improve the fit in that region (see Figure 13). The disk has an outer radius of 0.5 times the binary separation, and the distance obtained from the fit is 854 pc. The mass accretion rate obtained by integrating the luminosity of the disk is 3.7 × 10⁻⁹ M_⊙ yr⁻¹, and the heated WD luminosity is of the same order.

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The fit is improved further by increasing the WD temperature to 50,000 K and the isothermal disk temperature to 12,000 K; however, the distance increases to 1137 pc. This model is shown in Figure 14. The disk-integrated luminosity gives a mass accretion rate $\dot{M}=5\times {10}^{-9}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$, and here, too, the WD luminosity is of the same order.

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For the IUE spectrum, we find that for the model to fit the spectrum, the temperature of the disk has to be decreased as the temperature of the WD increases. The best fit (Figure 15) is obtained for a disk with a temperature of 10,000 K combined with a 40,000 K WD. This model also gives a distance of the order of 650 pc for a an outer disk radius ∼0.6a. The integrated luminosity of the disk gives a mass accretion rate of 2.5 × 10⁻⁹ M_⊙ yr⁻¹, while the WD luminosity is about 10% less than that of the disk.

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The results of the analysis of the UV spectra of BZ Cam can be summarized as follows: a 0.4 M_⊙ WD with a temperature of 45,000 ± 5000 K, and a disk with a temperature of 11,000 ± 1000 K, extending to 0.5 ± 0.1 times the binary separation. Taking only the disk luminosity into consideration gives a mass accretion rate of the order of 2.5 × 10⁻⁹ M_⊙ to 5 × 10⁻⁹ M_⊙ yr⁻¹, and twice as large when also taking the WD luminosity contribution (assuming that the WD is heated due to the advection of energy from the inner disk). The WD contributes mainly to the FUSE spectral range while the cold disk contributes mainly to the IUE spectral range. The WD mass could be larger, i.e., 0.7 M_⊙, but its radius would have to be of the order of 20,000 km, i.e., twice as large as expected for such a mass (again, see Section 3.2 for details).

We assumed a 0.8 M_⊙ WD mass with a radius R_wd = 7000 km, which is consistent with a temperature of a WD at about 16,000 K. The inner radius of the disk, R₀, was placed at 1.0 R_wd, 1.2 R_wd, 1.5 R_wd, and 2.0 R_wd. The outer radius was placed at 36 R_wd ≈ a/3, where a is the binary separation of V592 Cas derived from Table 1. The outer region of the disk was allowed to take different temperatures: either the actual disk temperature, or 10,000 K, 11,000 K, or 12,000 K. The size of that outer region was also varied. Some preliminary modelings were carried out using a grid of standard disk models to obtain an order of estimate of the mass accretion rate, which gave $\dot{M}\sim {10}^{-8}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$. Modified disk models were then computed within the vicinity of this mass accretion rate assuming an inclination angle of 30°.

The models were chosen to provide the best fit and the correct distance (≈350 pc) to the source. The best fit is for a modified disk with a mass accretion rate of $\dot{M}={10}^{-8.2}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$ (∼6.3 × 10⁻⁹ M_⊙ yr⁻¹), an inner disk radius placed at R₀ = 1.2 R_wd, an outer disk radius at 30 R₀, and an isothermal outer disk between r = 20 R₀ and r = 30R₀ with a temperature T = 12,000 K. A 45,000 K WD contribution was added to the model for completeness (Hoard et al. 2009) but contributed only a few percent to the flux and did not affect the model. Consequently, the temperature of the WD could not be assessed from our modeling. The fit is presented in Figure 16 in the FUSE spectral range and in Figure 17 in the IUE SWP+LWP spectral range.

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For a 16,000 K WD with a radius of 7000 km, the modified disk model, with an 8400 km inner radius, starts only at 1.2 R_wd, and for a 45,000 K WD with a radius of 7450 km, the disk starts at 1.13 R_wd. In either case, the size of the BL would be relatively narrow (∼0.1–0.2 R_wd), and this modified disk model is very similar to a standard disk model in its inner region and differs mainly in the assumption of an outer isothermal region. V592 Cas is not as extreme a case as BZ Cam or MV Lyr.