accretion disk in the massive binary ry scuti - IOPscience

The Astronomical Journal, 136:767–772, 2008 August c 2008. The American Astronomical Society. All rights reserved. Printed in the U.S.A.


doi:10.1088/0004-6256/136/2/767

ACCRETION DISK IN THE MASSIVE BINARY RY SCUTI G. Djuraˇsevi´c1,2 , I. Vince1 , and O. Atanackovi´c3

1

Astronomical Observatory, Volgina 7, 11060 Belgrade 38, Serbia Isaac Newton Institute of Chile, Yugoslavia Branch, Volgina 7, 11060 Belgrade 38, Serbia Department of Astronomy, Faculty of Mathematics, University of Belgrade, Studentski trg 16, 11000 Belgrade, Serbia Received 2008 April 8; accepted 2008 June 1; published 2008 July 11 2

3

ABSTRACT The UBVR light curves of the massive eclipsing binary RY Sct, obtained at the Maidanak Observatory from 1979 to 1992, have been re-analyzed in order to prove the hypothesis of the presence of an accretion disk in the system. This hypothesis is supported by a new spectroscopic study of Grundstrom et al., and by a specific light-curve shape exhibiting a slight asymmetry around the secondary minima and a small difference in the height of the successive maxima. The light-curve analysis was performed using a Roche model of a binary containing a geometrically and optically thick accretion disk around the more massive primary star. By solving the inverse problem, the orbital elements and the physical parameters of the system components and of the accretion disk were estimated for all the individual UBVR light curves. The model gives a consistent solution for the RY Sct binary system and supports the hypothesis of the existence of an optically thick disk around the massive component. Our results suggest a mass exchange between the components and a mass loss from the system. This could be considered as a possible mechanism of the formation of the accretion disk around the more massive component and of the circumbinary envelope of toroidal form in the orbital plane of the system. Key words: binaries: close – binaries: eclipsing – stars: early-type – stars: individual (RY Sct)

explain the current spectroscopic and photometric observations of the system. The orbital phases are calculated by using the Ciatti et al. (1980) ephemeris formulae

1. INTRODUCTION In a recent spectroscopic study Grundstrom et al. (2007) demonstrated that the early-type eclipsing binary RY Sct (HD 169515) is a system with orbital period P ∼ 11d.125, consisting of a supergiant (secondary, less massive star; O9.7 Ibpe spectral class) and a more massive primary star with associated spectrum of class B0.5 I. Although the binary has a single-line spectrum, using a tomographic reconstruction technique these authors managed to separate the individual spectra of the components. They found that the B0.5 I spectrum was actually that of the photosphere of an accretion disk surrounding the massive companion. The optically and geometrically thick accretion disk, lying in the orbital plane of the system, obscures a great part of the massive companion from the observer due to a small orbital inclination. For this reason, it is difficult to separate the spectrum of the primary massive component from the composite spectrum of the system. The main goal of this work is to test photometrically the accretion disk model of the massive interacting binary system RY Scuti, as proposed by Grundstrom et al. (2007) on the basis of spectroscopic study, as well as to estimate the system’s orbital and physical parameters.

Min I = 2443,342.42 + 11d.12471 × E. The maxima of the light curve in quadratures slightly differ and the falling branches of principal minima are steeper than the rising ones. The same situation is seen with the secondary minima. Djuraˇsevi´c et al. (2001) modeled the system assuming circular orbits and using a Roche model with spots on the components. The best fit involved high overcontact configuration including two bright spots on the more massive star. One of them, located in the neck zone connecting the stars, indicates a significant mass and energy transfer from the secondary to the primary component. The other bright spot, situated on the opposite side of the more massive star near the external Lagrange point L3 , is probably the source of the intensive gas outflow. In this way, the formation of the circumstellar nebula around the system can be explained. Djuraˇsevi´c et al. (2001) pointed out that the inner surface of a thermally emitting circumstellar dust torus, seen edge on, can be ionized by the central star. The recent spectroscopic study of the RY Sct system (Grundstrom et al. 2007) essentially supported the existence of a circumbinary torus, but at the same time ruled out the overcontact model. According to the overcontact Roche model (Djuraˇsevi´c et al. 2001), the absolute magnitudes of the components are nearly equal, so the spectra of both components should be clearly visible in the composite spectra of the system. However, Grundstrom et al. (2007) did not find any spectral features originating from the more massive primary star. Thus, instead of an overcontact configuration the spectroscopic study strongly suggests the existence of a semidetached system with an accretion disk surrounding the smaller, more massive star.

2. OBSERVATIONS AND MODEL The photoelectric UBVR observations of RY Sct were obtained at Maidanak Observatory (Uzbekistan) during 1979– 1992 (Djuraˇsevi´c et al. 2001) by using telescopes with apertures of 0.4 m and 0.6 m and a one-channel photometer with standard UBVR filters (λU = 3600 Å, λB = 4400 Å, λV = 5500 Å, and λR = 7000 Å). The light curves were well defined (see Figure 1), with orbital phases covered with a total of 435(U ), 484(B), 490(V ), and 471(R) individual observations. This provided a good basis to test different models of the system by the use of light-curve analysis and to choose the optimal model that can 767

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According to the recent data it is plausible to assume that the binary system RY Sct actually consists of at least four different constituents: the primary and secondary star, the accretion disk (with active bright regions), and a toroidal emission nebula. The presence of the last constituent is indicated by observations of spectral lines Hα , Si iii 9532 Å, and of a radio continuum emission at 15 GHz (Smith et al. 2001). Assuming that the toroidal nebula has a negligible influence on the light curve in the visible spectral region, in our Roche model for UBVR lightcurve analysis we only considered the first three constituents. In this way we reduced the number of model parameters, which is very important for a successful solution of the inverse problem. Still, we should keep in mind that we have a very complex model described by a large number of parameters. Here, we re-analyze the UBVR photometric observations applying the new Roche model with an optically thick accretion disk around the more massive component of RY Sct, formed by the gas stream flowing from the Roche lobe filling the secondary supergiant. The basic elements of the model with a plane-parallel accretion disk, together with the light-curve synthesis procedure, are given in Djuraˇsevi´c (1992, 1996). The disk edge is approximated by a cylindrical surface. In the current, modified version of the code, the disk thickness changes linearly with radial distance. The conically shaped disk (which can be concave, convex, or plane parallel as a special transition case) is introduced as a generalization of the simple concave disk model suggested by Zola (1991) and Zola & Ogloza (2001). The conical form of the disk surface is described by the disk radius (Rd ), and by the disk thicknesses at the edge (de ) and at the center of the disk (dc ). This is a rough approximation of the disk shape obtained by the current hydrodynamical modeling of mass transfer in close binary systems; see, e.g., Bisikalo et al. (2000), Harmanec et al. (2002), and Nazarenko & Glazunova (2003, 2006a, 2006b). The disk’s edge cylindrical surface is characterized by the temperature Td . Radiation from conical surface of the disk depends on the radial temperature profile. In this modified code, a variety of simple power-law radial temperature distribution profiles of the disk have been applied:  − 14  14  R∗ R∗ T (r) = Td , 1− 1− r Rd
aT Rd T (r) = Td , r aT 
r − R∗ , T (r) = Td + (T∗ − Td ) 1 − Rd − R∗   
r − R∗ aT . T (r) = Td + (T∗ − Td ) 1 − Rd − R∗


Rd r

aT 



(1)

(2) (3) (4)

Equation (1) is described in detail in Djuraˇsevi´c (1996) and Djuraˇsevi´c et al. (2005). In the first two equations, for the α-disk radial temperature distribution, the value of exponent aT shows how close is the radial temperature profile to the steady-state configuration (aT = 0.75). Equations (3) and (4) are obtained by modifying the distribution proposed by Zola (1991). We assumed that the disk was in physical and thermal contact with the central star. So the inner radius and temperature of the disk were set equal to the radius and temperature of the star (R∗ , T∗ ). The temperature of the disk’s edge (Td ) and the radial temperature distribution exponent (aT ) were taken to be free

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parameters. Besides, the radii of the star (R∗ ) and the accretion disk (Rd ) also appear as parameters. On the disk’s lateral side, in the zone where the gas stream falls onto the disk, the temperature is increased and the so-called hot spot is produced. The hot spot deforms the light curves, making them asymmetric. If we assume that the hot spot is located on the lateral side of the disk at the place where the line passing through the L1 point is a tangent to the disk, the disk dimensions uniquely determine the longitude of this active region. Although this approximation reduces the number of free parameters, in the code we have the longitude of the hot-spot region as a free parameter as an option. Moreover, due to the fall of an intensive gas stream, the disk surface in the zone of the hot spot can be deformed, producing a local concentration of radiation, which deviates from the uniform azimuthal distribution. In the code, this effect is described by the angle θrad between the line perpendicular to the local disk-edge surface and the direction of the hot-spot maximum radiation. The quality of the observed fit can be additionally improved by introducing a second active bright region, which is generally located on the opposite side of the disk. This active region could be interpreted as the effect of a disk-shaped deviation from the circular one, i.e., as a hump on the disk edge from which part of the disk material probably leaves the system. An additional interpretation of the bright-spot nature can be found in recent gasodynamical studies, e.g., Heemskerk (1994), Bisikalo et al. (1998, 2000), Harmanec et al. (2002), Nazarenko & Glazunova (2003, 2006a, 2006b), showing very complicated gas flows in the system and the complex structure of the disk. It is reasonable to expect that part of the gas stream and the disk material from this active region leave the system into the surrounding space, possibly forming a kind of circumbinary envelope. These active features modify the azimuthal distribution of the radiation coming from the accretion disk and can explain the asymmetry of the light curves. 3. ANALYSIS AND RESULTS The light-curve analysis was made by applying the inverseproblem method (Djuraˇsevi´c 1992) based on a simplex algorithm to estimate the optimal system parameters yielding the best fit of the observations. In order to ensure the convergence of the iterative process of the inverse-problem solution, it is very important to restrict the number of model free parameters. This can be done by fixing some of them in advance on the basis of independent information or an acceptable approximation of the RY Scuti elements. Consequently, we assumed the following. 1. The mass ratio is fixed to q = m1 /m2 = 4.2 based on the radial velocity study by Grundstrom et al. (2007). The subscripts (1, 2) refer to the primary (more massive component surrounded by a disk) and the secondary (less massive Roche lobe filling component), respectively. 2. The temperature of the primary (gainer) component is fixed at the value T1 = 38,000 K, consistent with its spectral type (O7) (Lang 1992), which corresponds to a main-sequence star with a mass of 30 M , as obtained spectroscopically by Grundstrom et al. (2007). 3. The temperature of the secondary component is fixed at the value T2 = 30,000 K, which is consistent with its spectral type (O9.7 Ib). Note that the same value was used by Grundstrom et al. (2007) for their synthetic template spectrum calculation.

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4. According to the temperatures of the system components, we set the gravity-darkening coefficients and component’s albedos at their theoretical values (β1,2 = 0.25, A1,2 = 1.0), corresponding to von Zeipel’s law for fully radiating shells and complete reradiation. 5. The filling factor for the critical Roche lobe of the secondary star was set to F2 = 1.0, since we assumed that this component completely fills its Roche lobe and loses mass through the L1 Lagrange point. 6. We assumed that the rotation of the components is synchronous with their orbital revolution. Therefore, we adopted f1,2 = ω1,2 /ωK = 1.00, where f1,2 is the ratio of the angular rotation rate (ω1,2 ) to the Keplerian (ωK ) orbital rate. Instead of the rough first-order linear approximation, in our new code we applied the nonlinear limb-darkening approximation given by Claret (2000). For the effective wavelengths of the corresponding UBVR filters, the limb-darkening coefficients were interpolated from Claret’s tables according to the current values of the stellar effective temperature Teff and surface gravity log g in each iteration. More details concerning the interpolation procedure can be found in Djuraˇsevi´c et al. (2004). In order to explain the light-curve asymmetry we introduced a hot-spot region on the disk edge at the place where the gas stream from the secondary falls on the disk. The preliminary analysis showed that a hot-spot region cannot explain the light-curve asymmetry completely. By introducing an additional bright-spot region, larger in size and located on the nearly opposite side of the disk, the fit was much improved. The solution of the light-curve analysis, based on this model of RY Sct, is given in Table 1. The parameters are given in the first column, and the appropriate values for UBVR filters are given in the second, third, fourth, and fifth columns, respectively. The last column gives the mean values obtained from all four filters with the associated formal errors. These errors were estimated by combining the individual UBVR solutions and the results of numerous numerical tests obtained with a simplex algorithm, initialized by different input parameters. The first three rows of Table 1 present the number n of UBVR observations, the final sum of the squares of the residuals between observed (LCO) and synthetic (LCC) light curves n (Oi − Ci )2 , i=1

and the root mean square (rms) of the residuals 

n 2 i=1 (Oi − Ci ) σrms = , (n − 1) respectively. Apart from the geometrical and physical parameters, the table lists the characteristics of the spots located on the disk edge (temperature factor Ahs,bs = Ths,bs /Td , angular radius θhs,bs , and longitude λhs,bs ). The subscripts (hs, bs) refer to the hot and bright spots, respectively. In the code, longitudes of hot and bright spots (λhs,bs ) are measured clockwise (as viewed from the direction of the +Z-axis) from the +X-axis (the line connecting the stars’ centers) in the range 0◦ –360◦ . The inclination of the orbit was estimated to be i ∼ 82.◦ 4, so the observer can see the radiation originated mainly from the disk’s edge whereas the radiation from other (inner) parts of the disk is less pronounced. This fact makes the estimation

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of the temperature distribution along the disk’s radius difficult since its influence on the observed light curve is relatively weak. With the radial temperature distributions described by Equations (3) and (4) we obtain a good fit of the observations. With the other two radial temperature distributions, the convergence of the simplex minimization process is less efficient, producing larger discrepancies among individual UBVR solutions. Finally, we chose the results obtained with disk temperature distribution (3) as the optimal ones. By combining our photometric solutions with spectroscopic data (Grundstrom et al. 2007), we estimated the basic elements of the system, listed in the last 12 rows of Table 1. The RY Sct binary system consists of the more massive, primary component with a temperature of about 38,000 K. Its mass and radius are 30.0 M and 9.7 R , respectively. This component (surrounded by the accretion disk) eclipses the secondary (Roche lobe filling star) at the deeper (primary) minimum of the light curves. The secondary (donor) component, whose temperature is about 30,000 K, has a mass of 7.1 M and radius of 18.4 R . System separation is 69.9 R . The light-curve analysis shows that the accretion disk must be very large. The disk radius (Rd = 35.1 R ) is about 98% of the corresponding Roche lobe radius. The outer disk-edge thickness is de = 8.3 R and the central one is dc = 13.0 R . Therefore, the disk has a slightly convex shape. Because of this, and due to the high system inclination, the disk eclipses a great part of the radiation coming from the primary. The disk temperature increases from the edge (Td ∼ 23,200 K) to the center according to Equation (3). The disk temperature profile parameter is estimated to be aT ∼ 1.3. Consequently, the effective disk temperature is significantly higher. We note that in the case when the disk is oriented nearly edge on, the sensitivity of the inverseproblem solutions to the parameter aT is relatively weak and its error may be larger than the formal, tabulated one. It was shown that the light-curve asymmetry could be simulated by the presence of two active bright regions on the disk edge, located on nearly opposite longitudes. The temperature of the hot spot, located at longitude λhs ∼ 333◦ , at the place where the gas stream falls onto the disk, is approximately 73% higher than the disk-edge temperature. The direction of its maximum radiation is inclined by θrad ∼ 17◦ with respect to the line perpendicular to the local disk-edge surface. The larger bright spot (with temperature about 44% higher than the disk-edge temperature), contributing essentially to the light-curve asymmetry, is placed at longitude λbs ∼ 101◦ . This region can be interpreted as the zone in which the disk significantly deviates from the circular shape. It is plausible to expect that part of the disk material leaves the system from this region. Due to orbital motion, this material, together with mass loss through stellar wind, contributes to the circumbinary toroidal nebula of a very complex structure (Grundstrom et al. 2007) in the equatorial plane of the system. Following from the inverse-problem solutions for individual UBVR light curves, Figure 1 (left) presents the optimal fit of the observed light curves (LCO) with the synthetic ones (LCC). The light curves are normalized to the system brightness at the orbital phase 0.25. The view of RY Sct’s optimal model at orbital phases 0.25 and 0.75 is shown at the bottom. The right-hand column in Figure 1 shows the individual UBVR synthetic fluxes of the components (STAR1, STAR2, and DISK), normalized to the total synthetic flux of the system at orbital phase 0.25. It is evident that the main flux contribution comes from the Roche lobe filing secondary (almost 60% at the

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Table 1 Results of the Analysis of RY Sct Light Curves, Obtained by Solving the Inverse Problem for the Roche Model with Accretion Disk Around the More Massive Component Quantity

U-filter

B-filter

V-filter

R-filter

Mean

n 435 484 490 471 Σ(O − C)2 0.5777 0.4074 0.3115 0.2513 σ rms 0.0365 0.0293 0.0252 0.0231 i 82.6 82.3 82.5 82.4 82.4 ± 0.3 Fd 0.98 0.98 0.98 0.98 0.98 ± 0.02 Td 22590 23350 23440 23450 23200 ± 800 de 0.12 0.12 0.12 0.12 0.12 ± 0.01 dc 0.19 0.19 0.19 0.17 0.19 ± 0.02 F1 0.29 0.29 0.29 0.29 0.29 ± 0.01 Ahs = Ths /Td 1.73 1.73 1.74 1.73 1.73 ± 0.03 θ hs 22.3 24.1 24.6 24.0 23.8 ± 1.5 λhs 332.5 333.4 333.6 333.4 333.2 ± 2.0 Abs = Tbs /Td 1.43 1.43 1.44 1.45 1.44 ± 0.03 θ bs 47.4 46.4 42.0 46.4 45.6 ± 3.6 λbs 93.1 103.4 102.2 103.8 100.6 ± 7.5 −16.8 −17.6 −17.1 −17.7 −17.3 ± 1.5 θ rad aT 1.27 1.36 1.34 1.25 1.3 ± 0.1 Ω1 7.45 7.47 7.44 7.45 7.45 ± 0.02 Ω2 2.325 2.325 2.325 2.325 2.325 M1 (M ) 30.0 30.0 30.0 30.0 30.0 ± 2.1 M2 (M ) 7.14 7.14 7.14 7.14 7.1 ± 1.2 R 1 ( R ) 9.70 9.67 9.71 9.70 9.7 ± 0.2 R 2 ( R ) 18.39 18.39 18.39 18.39 18.4 ± 0.3 log g1 3.94 3.94 3.94 3.94 3.94 ± 0.03 log g2 2.76 2.76 2.76 2.76 2.76 ± 0.02 1 Mbol −8.33 −8.32 −8.33 −8.33 −8.3 ± 0.1 2 Mbol −8.69 −8.69 −8.69 −8.69 −8.7 ± 0.1 aorb (R ) 69.9 69.9 69.9 69.9 69.9 ± 2.3 Rd (R ) 35.11 35.09 35.08 35.09 35.1 ± 0.1 de (R ) 8.10 8.42 8.40 8.31 8.3 ± 0.3 dc (R ) 13.1 13.3 13.5 12.1 13.0 ± 0.9 Notes. Fixed parameters: q = m1 /m2 = 4.2, mass ratio of the components; T1 = 38,000 K, temperature of the more massive gainer star (surrounded by the accretion disk); T2 = 30,000 K, temperature of the less-massive Roche lobe filling star; F2 = 1.0, filling factor for the critical Roche lobe of the less-massive star; f 1 = f 2 = 1.00, nonsynchronous rotation coefficients; β1,2 = 0.25, gravity-darkening coefficients; A1,2 = 1.0, albedo coefficients of the components. n, number of observations; Σ(O − C)2 , final sum of squares of the residuals between observed (LCO) and synthetic (LCC) light curves; σrms , rms of the residuals; i, orbital inclination (in degrees); Fd = Rd /Ryc , disk dimension factor; Td , disk-edge temperature; de, dc, disk thicknesses (at the edge and the center of the disk, respectively) in the units of the distance between the components; F1 , filling factor for the critical Roche lobe of the more massive gainer star; Ahs,bs = Ths,bs /Td , hot and bright spots’ temperature coefficient; θhs,bs and λhs,bs , spots’ angular dimensions and longitudes (in degrees); θrad , angle between the line perpendicular to the local disk-edge surface and the direction of the hot-spot maximum radiation; aT , disk temperature distribution coefficient (see Equation (3)); Ω1,2 , dimensionless surface potentials of the primary and secondary; M1,2 [M ], R1,2 [R ], stellar masses and mean radii of stars in solar units; 1,2 log g1,2 , logarithm (base 10) of the system components’ effective gravity; Mbol , absolute bolometric magnitudes of the components; aorb (R ), Rd (R ), de (R ), dc (R ), orbital semimajor axis, disk radius, disk thicknesses at its edge and center, respectively, given in the solar radius units.

light-curve maxima) and that the contribution of the primary, mostly hidden by the disk, is practically negligible (about 4%). This can explain why there is no reliable identification of spectral features related to the primary component. Moreover, photometrically, the contribution of this star to the total system light can essentially be ignored. We obtained the fit of similar quality assuming that the central star is a compact body, completely hidden by the disk. In that case, the more massive component

of RY Sct could be interpreted as a collapsed star surrounded by an optically and geometrically thick disk. However, in accordance with evolutionary scenario described in Grundstrom et al. (2007), we accepted the solution with stellar central body of the finite radius (R1 = 9.7 R ), which is quite close to the value given by these authors. This model gives a slightly better fit of UBVR light curves than that with the collapsed central body.

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Figure 1. Left: UBVR observed (LCO) and synthetic (LCC) light curves of RY Sct; right: UBVR fluxes of components and the accretion disk; bottom: the view of the optimal model at orbital phases 0.25 and 0.75.

4. DISCUSSION AND CONCLUSION In this paper we modeled UBVR light curves of the eclipsing binary star RY Sct (Djuraˇsevi´c et al. 2001) using a new improved code for systems with an accretion disk, obtained by modification of the code described in Djuraˇsevi´c (1996) and Djuraˇsevi´c et al. (2005), and the inverse-problem method based on a simplex algorithm. The present analysis shows that the massive binary RY Sct is in the phase of an intensive

mass transfer from the Roche lobe filling secondary toward the primary, leading to the formation of an optically and geometrically thick accretion disk around the more massive primary star. This model fits the observations very well. The individual UBVR light-curve solutions are in good mutual agreement, indicating that the model can be used to explain the photometric behavior of RY Sct. Comparing our results with solutions obtained using the overcontact Roche model without a disk (Djuraˇsevi´c et al. 2001), it is evident that the disk model

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gives only a slightly better fit of the observations, but its main advantage is in the capacity to explain the basic spectroscopic characteristics of this binary system (Grundstrom et al. 2007). Hence, this paper offers an important validation of the basic model features presented by these authors. In contrast to the traditional models in which the hot-spot region is formed on the disk lateral side in the zone where the gas stream falls onto the disk, contemporary three-dimensional numerical simulations of gaseous flow structure show significant changes in the stream-disk morphology (Bisikalo et al. 1998). The stream deviates under the action of the gas in the circumbinary envelope, which leads to a shock-free (tangential) interaction between the stream and the edge of the forming accretion disk. As a consequence, the traditional hot spot does not exist. At the same time, part of the stream revolves around the accretor and interacts with the stream itself causing the formation of an extended shock wave, located outside the disk on the stream edge (the “hot line”). The hydrodynamical simulations of the mass flow in long-period interacting binaries like β Lyrae (Bisikalo et al. 2000) have shown a variety of structures such as accretion disk, circumstellar envelope, “hot line,” and cool stream halo. Since our solutions show that the disk fills the Roche lobe almost completely (98%), the secondary is close to the edge of the disk. Therefore, tidal forces on the disk will be large and, as is well known from hydrodynamical calculations (Heemskerk 1994), we can expect a spiral-shaped tidal shockwave in the disk. We can assume that in a real situation a part of the gas stream and the material from the disk leave the system, possibly forming a kind of circumbinary envelope, which could also influence the light-curve shape to some degree. Therefore, the real gas dynamics in the system might be more complicated than the simplified model we employed here. However, the present study, based on the available photometric and spectroscopic data, shows that our simple model is able to describe the light-curve features (asymmetry, shape, etc.) very well, i.e., the model can successfully explain the basic spectral and photometric characteristics of the system. The light-curve analysis confirms the main results obtained spectroscopically by Grundstrom et al. (2007). We point out that we derived larger inclination (i ∼ 82.◦ 4) of the orbital plane than that obtained by the geometrical shape of the outer nebula (i ∼ 75.◦ 6) found by Smith et al. (1999). Our study shows that it is not possible to obtain a good fit of the light curves with inclination i ∼ 75.◦ 6, neither with the present disk model, nor using some alternative ones (see Antokhina & Cherepashchuk 1988, Antokhina & Kumsiashvili

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1999, and Djuraˇsevi´c et al. 2001, who give the inclination in the interval 82◦ –85◦ ). For the discrepancy between the latter orbital inclination results (including the solution derived here) and that for the circumbinary gas (Smith et al. 1999) we have no satisfactory explanation. The present accretion disk model can be improved by using new photometric and spectroscopic observations and by analyzing the orbital period changes. Such an analysis would be very useful as it could provide us with estimates of mass transfer and mass-loss ratios. We are grateful to M. Zakirov and M. Eshankulova for providing the photometric data and to D. Gies for valuable comments and constructive suggestions that helped us to improve the paper. The authors acknowledge the support by the Ministry of Sciences of Republic of Serbia through the project “Stellar and Solar Physics” (No. 146003). REFERENCES Antokhina, E. A., & Cherepashchuk, A. M. 1988, Sov. Astron. Lett., 14(2), 105 Antokhina, E. A., & Kumsiashvili, M. I. 1999, Astron. Lett., 25(10), 662 Bisikalo, D. V., Boyarchuk, A. A., Chechetkin, V. M., Kuznetsov, O. A., & Molteni, D. 1998, MNRAS, 300, 39 Bisikalo, D. V., Harmanec, P., Boyarchuk, A. A., Kuznetsov, O. A., & Hadrava, P. 2000, A&A, 353, 1009 Ciatti, F., Mammano, A., Margoni, R., Milano, L., Vittone, A., & Strazzulla, G. 1980, A&AS, 41, 143 Claret, A. 2000, A&A, 363, 1081 Djuraˇsevi´c, G. 1992, Ap&SS, 196, 267 Djuraˇsevi´c, G. 1996, Ap&SS, 240, 317 Djuraˇsevi´c, G., Albayrak, B., Selam, S. O., Erkapi´c, S., & Senavci, H. V. 2004, New Astron., 9, 425 Djuraˇsevi´c, G., Rovithis-Livaniou, H., Rovithis, P., Borkovits, T., & Bir´o, I. B. 2005, New Astron., 10(6), 517 Djuraˇsevi´c, G., Zakirov, M., Eshankulova, M., & Erkapi´c, S. 2001, A&A, 374, 638 Grundstrom, E. D., Gies, D. R., Hillwig, T. S., McSwain, M. V., Smith, N., Gehrz, R. D., Stahl, O., & Kaufer, A. 2007, ApJ, 667, 505 Harmanec, P., Bisikalo, D. V., Boyarchuk, A. A., & Kuznetsov, O. A. 2002, A&A, 396, 937 Heemskerk, M. H. M. 1994, A&A, 288, 807 Lang, K. R. 1992, Astrophysical Data: Planets and Stars (Berlin: Springer) Nazarenko, V. V., & Glazunova, L. V. 2003, Astron. Rep., 47(12), 1013 Nazarenko, V. V., & Glazunova, L. V. 2006a, Astron. Rep., 50(5), 369 Nazarenko, V. V., & Glazunova, L. V. 2006b, Astron. Rep., 50(5), 380 Smith, N., Gehrz, R. D., & Goss, W. M. 2001, AJ, 122, 2700 Smith, N., Gehrz, R. D., Humphreys, R. M., Davidson, K., Jones, T. J., & Krautter, J. 1999, AJ, 118, 960 Zola, S. 1991, Acta Astron., 41, 213 Zola, S., & Ogloza, W. 2001, A&A, 368, 932