Dwarf Nova Outbursts. I. The UV Delay

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1 ACTA ASTRONOMICA Vol. 48 (1998) pp Dwarf Nova Outbursts. I. The UV Delay by Józef I. S m a k N. Copernicus Astronomical Center, Polish Academy of Sciences, ul. Bartycka 18, Warsaw, Poland jis@camk.edu.pl Received November 10, 1998 ABSTRACT Models of dwarf nova outbursts are calculated for a range of relevant parameters: the masses of the components which determine the orbital period and the radius of the disk, the mass transfer rate, and the viscosity parameter. The visual (5500 Å) and ultraviolet (1200 Å) light curves are calculated using black body and Kurucz fluxes. The resulting UV delays show strong dependence on model parameters and, in particular, confirm the existence of a dichotomy between Type A and Type B outbursts. The model predicted UV delays agree quite well with observations, with the "best-fit" for hot between 0.1 and 0.2. Earlier, negative conclusions and the controversy around the "problem of UV delay" resulted primarily from incorrect outer boundary conditions and other model simplifications. Key words: binaries: close novae, cataclysmic variables 1. Introduction Following the pioneering work by Meyer and Meyer-Hofmeister (1981), numerous papers have been published by many authors, presenting the results of calculations based on accretion disk models, including the effects of their thermal instability, and reproducing successfully main features of dwarf nova outbursts (see recent reviews by Cannizzo 1993 and Osaki 1996). Early models predicted two types of outbursts (Smak 1984): Type A (or outside-in) when the instability begins in the outer parts of the disk and propagates inward, and Type B (or inside-out) when the instability first occurs in the inner parts. Nearly all observable characteristics of Type A and Type B outbursts differ considerably. It was found, in particular, that the initial rise does not occur simultaneously at all wavelengths and that this strongly depends on the type of outburst. Type A outburst begins with the heating and brightening of the outer parts of disk. The increasing viscosity causes the material to flow inward and this prevents any

2 678 A. A. excessive heating at this phase. Due to the relatively large area involved, however, this initial, moderate heating is already sufficient to produce a significant increase in the visual flux. At shorter wavelengths, no comparable brightening occurs until the accreting material reaches inner parts of the disk what results in a major increase of their temperature. The effects can be best seen when comparing the visual and far ultraviolet light curves, hence the term: the UV delay. In Type B, the situation is more complex. At small radii the thermal time scale and the viscous time scale can be comparable. Consequently, the two competing effects: the local heating and the propagation of thermal instability, enlarging the radiating area, occur almost simultaneously. And it is primarily due to the second effect that the visual flux rises earlier and the UV delay (relatively smaller, however, than in Type A) is also produced. Observational evidence available in mid 80s appeared to confirm these predictions: examples of relatively large and relatively small UV delays could, indeed, be identified with respectively Type A and Type B outbursts (Smak 1984, 1987, Fig. 1). At about the same time, however, several papers appeared (e.g., Pringle et al. 1986, Cannizzo and Kenyon 1987, Meyer and Meyer-Hofmeister 1989) with clearly negative conclusions. According to them, the UV delays predicted by disk instability models of dwarf nova outbursts disagreed with observations, being significantly too small. Notable exception in this respect were two papers by Mineshige (1986, 1988) who found that model predicted UV delays do agree with observations but his conclusions had no broader impact on public opinion. The existence of the "problem of UV delay" was generally accepted and numerous papers were soon to be published, presenting its possible explanations (none of them fully successful in this respect). The purpose of this paper is to present in Section 2 models of dwarf nova outburst, calculated for a broad range of relevant parameters,to show in Section 3 that model predicted UV delays agree well with observations and, finally, to discuss in Section 4 major causes which contributed to the controversy and confusion surrounding the "problem of UV delay" Disk Models 2. Model Calculations Time dependent disk models were calculated in the same way and under the same assumptions as in Smak (1984). Consequently, they are similar or nearly identical with those presented and discussed in that paper. The input parameters are: the masses of the components, M 1 and M 2, which determine the orbital period and the mean radius of the disk, and the mass transfer rate, _M. The? T e relation used in the calculations (Smak 1964, Fig. 1) is a standard relation, arbitrarily modified at lower temperatures in a way which is essentially identical with the more common practice of adopting a lower value of cool for the lower

3 Vol branch of the? T e relation. Consequently, there is one free parameter, namely the value of on the hot branch, i.e., = hot. T a b l e 1 Model Data M 1 M 2 log _ M P orb hr d i t W d d Models are listed in Table 1, with their numbers so chosen as to indicate some of their properties: Models were calculated with = 0:1, while models with = 0:2. Models with odd numbers produce Type A outbursts, while those with even numbers Type B outbursts. The orbital periods listed in the Table are given in days and the mean disk radii in cm.

4 680 A. A Disk Light Curves Monochromatic disk light curves were calculated at two wavelengths: = 1200 Å and 5500 Å. The choice of the ultraviolet wavelength was dictated by the fact that model results can be compared with the shortest wavelength data from the IUE, as well as with the 1050 Å data from Voyager. On the other hand 1200 Å may appear too close to the position of the L line. In the spectra of dwarf novae, however, this line is present (in absorption) only in ultrashort period systems at quiescence (due to the white dwarf). In other systems and in all systems during outbursts it has never been detected (neither in emission nor in absorption). The fluxes were calculated in two ways: in the black body approximation (BB) and using the flux vs. T e calibration based on model stellar atmospheres published by Kurucz (1979). Since Kurucz fluxes at 1200 Å are affected by the L absorption line, the effective continuum flux at this wavelength was interpolated between 1112 Å and 1312Å. In the context of present applications the difference between the black body and Kurucz fluxes shows up in the ratio of fluxes at the two wavelengths R = f (1200 Å)=f (5500 Å) and its dependence on temperature. At low temperatures R(Kurucz) is much lower than R(BB); they become equal at about 9000 K, and at higher temperatures R(Kurucz) is much larger. Consequently, for hot disks the Kurucz fluxes at 1200 Å are expected to be relatively higher (with the resulting UV delay being smaller) but for cooler disks the effect is likely to be smaller. One should note here that the absolute fluxes from the disk depend on inclination, and that those calculated here are the mean fluxes corresponding to an inclination of, roughly, about 60. Obviously, the shapes of the disk light curves and, in particular, the UV delays determined from them, do not depend on inclination. The effect may become relevant, however, in the case of system light curves (Section 2.3), including contributions from the two stars and from the hot spot. We shall return to this problem in Section System Light Curves For comparison with observations we must have light curves including contributions from the disk as well as from the remaining components of a given system. Those system light curves were also calculated, using black body and Kurucz fluxes as follows. The white dwarf. Its temperature at quiescence was calculated as a function of the orbital period from a correlation shown by data for white dwarfs in dwarf novae (Sion 1991, Fig. 2). During outburst the white dwarf is heated by radiation from the boundary layer. This was taken into account by assuming that one-half of the total flux from the boundary layer is intercepted by the surface of the white dwarf (the other half being radiated away in the form of X-rays). The temperature of the white dwarf was then calculated by adding one-half of the flux from the boundary layer to its quiescent flux. The implicit assumption of instantaneous heating and

5 Vol cooling, which is clearly incorrect in the case of cooling, will affect the resulting light curves only during the final decline and therefore is of no consequence for the UV delay during rise. The secondary. Fluxes from the secondary component were calculated as a function of its mass using the temperature calibration based on data for lower main sequence stars from Popper (1980). The hot spot. Fluxes from the hot spot were determined by calculating its bolometric luminosity and assuming that its effective temperature is K. Fig. 1. Examples of system light curves (normalized to their maximum level). Solid lines are the visual (5500 Å) light curves calculated with Kurucz fluxes. Broken lines the ultraviolet (1200 Å) light curves calculated with Kurucz fluxes while dotted lines with black body fluxes. Note that time is expressed in 10 5 seconds (not in days used everywhere else in this paper) and that the scale of the upper and lower plots is different. Examples of system light curves, presented in Fig. 1, illustrate the effects of the orbital period (or the radius of the disk) and of the type of outburst. Since the visual light curves calculated with black body and Kurucz fluxes are practically identical, only those based on Kurucz fluxes are shown in Fig. 1. In the case of ultraviolet the situation is different: with the black body fluxes being systematically lower, the corresponding light curves are systematically delayed with respect to those based

6 682 A. A. on Kurucz fluxes. Note that in the case of Type A outbursts this effect is smaller than in Type B (we shall return to this point in Section 4.1) The UV Delay Model light curves at 1200 Å and 5500 Å were now used to determine the UV delay (t), defined arbitrarily at a level log f =?1:0, i.e., 2.5 mag below maximum, corresponding to early rise. The width of outburst (W 1 ), to be used in further discussion, was also determined from visual light curves at a level 1 mag below maximum. The results are listed in Table 1 where the first line for each model gives values of t and W 1 (in days) based on Kurucz fluxes, the second on black body fluxes. In addition to system light curves, values of t were also determined from disk light curves. A comparison of the two sets of t shows that (with two exceptions discussed below) they differ (in plus or in minus) by less then percent. This is a direct consequence of the fact that contributions from the three additional sources are significant only at quiescence but negligible during outburst. This also implies (and calculations confirm it) that, as long as the contribution from the disk is dominant (which is the case at moderate and low inclinations), including the effects of its inclination (see Section 2.2) would not change the picture in any significant way. Situation would be more complicated, however, for inclinations close to 90 but, fortunately, observational data do not include such cases. In Models 18 and 28 the contribution from the secondary component to the visual flux is so large that the flux at quiescence is already above the log f =?1:0 level (and the resulting outburst amplitude is less than 2.5 mag). Consequently, values of t were determined at a somewhat higher level of log f =?0:8. For comparison, listed in Table 1 as Models 18d and 28d, are values of t based on disk light curves. The two sets differ in this case significantly. As we shall see below (Figs. 2 and 3), however, this difference is of no importance for our main conclusions Model Limitations Our models did not include one important effect: heating of the secondary component by the boundary layer during outbursts, resulting in the enhanced mass outflow. As it begins when the accretion wave reaches the white dwarf, it does not affect the early phase of an outburst. Therefore neglecting this effect was, fortunately, of no consequence for the UV delay during rise. During later phases, however, it plays an important role and its neglect has important consequences for model light curves. This is the most likely explanation of the fact that all models published so far failed to reproduce the extended duration (or flat-topped shape) of many observed outbursts. For this reason the widths of outbursts determined from model light curves should be treated with caution.

7 Vol Fig. 2. Comparison of UV delays based on model light curves calculated with Kurucz fluxes with observations. Observational data are shown as asterisks (multiple points for the same object being connected with vertical lines). Model data are plotted using the following symbols: squares models with = 0:1, triangles models with = 0:2, filled symbols models with Type A outbursts, and open symbols models with Type B outbursts. Double symbols at P orb = 0:31 d represent models 18/18d and 28/28d (see Section 2.4).

8 684 A. A. Fig. 3. Same as in Fig. 2 for black body fluxes.

9 Vol Analytical Considerations In the case of Type A outbursts, assuming that the UV delay represents travel time of the accretion wave, it is possible at least in principle to describe it analytically. In the -disk approximation the radial velocity of the accreting matter is V 3 r = 2 GM1 1=2 z0 2 (1) R R where z 0 =R depends on a number of parameters involved, in particular on. Adopting this dependence in the form (z 0 =R)?0:15 (Smak 1992), neglecting its (weak) dependence on R, and integrating Eq. (1), we get for the travel time between R d and the inner radius R in t(r d ; R in ) 0:7 (R d 3=2? R in 3=2 ): (2) To make a comparison, we use pairs of models with different values of and hr d i. In the case of, from two such pairs Models 11/21 and 13/23 we get t( = 0:1)=t( = 0:2) = 1:59? 1:74 which compares favorably with 1.62 resulting from Eq. (2). In the case of hr d i, another two pairs Models 13/11 and 23/21 give t(hr d i = 4:2)=t(hR d i = 1:9) = 3:21? 3:52 to be compared with 3.29 resulting from Eq. (2) Comparison with Hameury et al. (1998b) The only other models, calculated with correct outer boundary conditions (see Section 4 for discussion of this problem), are those by Ichikawa and Osaki (1992) and by Hameury et al. (1998a, 1998b). Of those three papers only the last one presents results concerning the UV delay. In fact, Hameury et al. (1998b) discuss the EUV delay which refers to the extreme UV flux coming primarily from the boundary layer and, therefore, it is assumed to be proportional to the central accretion rate M _ wd. Obviously then, this EUV delay must be systematically longer than the "standard" UV delay, considered in this and other papers. The difference, however, can be expected to be rather small (on account of the relatively very short travel time at small radii; see Eq. 2), and this is actually confirmed by the nearly identical delays obtained for SS Cyg from the EUVE (Mauche 1996) and Voyager (Cannizzo et al. 1986) data (see next Section). In what follows, we shall therefore compare our t with t EUV from models by Hameury et al. (1998b). Their first model (Hameury et al. 1998b, Figs. 4 and 6) was calculated with the following set of parameters: = 0:2, M 1 = 0:6M, hr d i = , This model is nearly identical with our Model 21, with = 0:2 and hr d i 1: (Note that in choosing pairs of comparable models we take into account primarily the strong, obvious dependence of the UV delay on and on the radius of the disk (see Eq. 2), with other parameters being less important). To

10 686 A. A. determine the EUV delay in the same way as in Section 2.4, we assume that m x = 27? log M _ wd, plotted in Fig. 6 of Hameury et al. (1998b), was chosen in such a way, as to make log M _ wd (max) coincide with V (max) 9:5, i.e., that log M _ wd (max) 17:5. The log M _ wd curve can then be redrawn on a magnitude scale (or the V curve on a log scale) and used to determine the EUV delay at the log f =?1:0 and log M _ wd =?1:0 levels. For the case "without illumination" we get t EUV = 6:5 hrs = 0:27 d, in perfect agreement with t values obtained from Model 21. The second model of Hameury et al. (1998b, Figs. 7 and 8) was calculated with the following set of parameters: = 0:1, M 1 = 1:2M, hr d i = The type of outburst was not specified, but two independent evidences help to make unambiguous identification. The first is the rounded shape of the outburst (Fig. 7) with a slow rise, characteristic for Type B. The second is based on the delay between the initial rise in M _ wd and the initial rise of the visual flux. In the first model this delay was about 4 hours (Fig. 6). Should the outburst produced in the second model be also of Type A, then we would expect (from Eq. 2) the delay to be 1:6 longer on account of a smaller and 2:8 longer on account of a larger R d, the resulting expected value being about 18 hours. Instead the delay is only about 5 or 8 hours (dotted and broken lines in Fig. 8), and this implies that it must have been of Type B. If so, the second model should be compared with our Model 14, with = 0:1 and hr d i 4: We assume that, in order to make log M _ wd (max) coincide with V (max) 11:0, m x must have been defined in this case with a different constant (29 instead of 27), so that log M _ wd (max) 18:0. Repeating the procedure described above for the dotted and broken curves corresponding to the illumination case we get t EUV = 21? 22 hrs 0:9 d. This is nearly identical with the larger of the two values of t obtained from Model 14 (corresponding to the black body case). Unfortunately, no data are available for the case "without illumination" Observational Data 3. Comparison with Observations Observational data on UV delays are listed in Table 2 where P orb, t, and W 1 are given in days. Values of t were determined, whenever possible, during early rise and W 1 at a level 1 mag below visual maximum. In addition to observations discussed earlier (Smak 1987), Table 2 includes a few more recent determinations. Among them are the EUVE data for SS Cyg (Mauche 1996) which in spite of a much shorter wavelength do not differ from the Voyager data (Cannizzo et al. 1986), and the broad band (1250? 1950 Å) IUE data for VW Hyi (Wheatley et al. 1996) which agree with earlier IUE data at 1200 Å and with the Voyager data (Polidan and Holberg 1987). Concerning the accuracy, one should note that data listed in Table 2 are highly inhomogeneous in this respect: values of t based on

11 Vol well covered ultraviolet light curves from Voyager and EUVE are very accurate while some of those based on fragmentary IUE data (marked in Table 2 with colon or double colon) are uncertain or very uncertain. T a b l e 2 Observations Star P orb t W 1 Type UV Ref. RX And : 6.4 B IUE 7 SS Cyg B Voyager EUVE 3 AH Her B IUE 7 VW Hyi :: 2.0 A IUE 2 0.6: 1.7 IUE Voyager IUE 8 WX Hyi :: 2.0: A IUE 2 CN Ori B IUE 5 References to Table 2: (1) Cannizzo et al. 1986, (2) Hassall et al. 1985, (3) Mauche 1996, (4) Polidan and Holberg 1987, (5) Pringle et al. 1986, (6) Schwarzenberg-Czerny et al. 1985, (7) Verbunt et al. 1984, (8) Wheatley et al Models vs. Observations Ideally, taking into account the fact that the UV delay t strongly depends on the radius of the disk R d, the comparison of model predictions with observations should involve normalized values of t=r d. This cannot be done, however, since the radii of disks in systems listed in Table 2 are not known with sufficient accuracy. Fortunately, the radius of the disk depends on other system parameters in such a way that it is closely correlated with the orbital period. The comparison of model predictions with observations is presented in Figs. 2 and 3 with their top panels showing t values plotted against the orbital period and the other two panels t normalized with the width of outburst (which also depends strongly on R d ; but see comments in Section 2.5) and with the orbital period. On the basis of this comparison we can make the following conclusions. (1) Model predictions agree with observations quite well. The agreement is somewhat poorer only for the black body fluxes in the case of t=w 1 (but see comments on W 1 in Section 2.5).

12 688 A. A. (2) The dichotomy between Type A and Type B outbursts, predicted by models, is confirmed. Consequently outburst types can be unambiguously assigned to all observed cases (column "Type" in Table 2); they all agree with earlier identifications (Smak 1987). (3) In the case of models using Kurucz fluxes the best "fit" could be obtained for somewhere between 0.1 and 0.2. Models with black body fluxes would suggest close to 0.2. This is, roughly, consistent with an earlier estimate of hot 0:2 (Smak 1984) obtained from the slopes of the declining parts of outburst light curves. It would be too risky, however, to conclude at this point that black body fluxes are "better" than Kurucz fluxes The UV Delay during Decline Earlier analysis of the observational data suggested that, in addition to the UV delay during rise, similar delay occurs during decline, and that the two delays are correlated (Smak 1987, Fig. 1). Model light curves (Fig. 1; see also Fig. 3 in Cannizzo et al. 1986) show that the situation during decline is more complex: the initial UV delay is replaced by the "UV advance", with a cross-over occurring half-way between maximum and quiescence. Such a behavior can indeed be seen in Voyager light curves of VW Hyi (Polidan and Holberg 1987, Fig. 1), and less clearly of SS Cyg (Cannizzo et al. 1986, Fig. 1); note that they represent Type A and Type B, respectively. In view of model limitations discussed in Section 2.5, any further discussion appears premature. Problem deserves further attention and better models. 4. Discussion 4.1. The Outer Boundary Conditions As discussed in the Introduction, the very existence of UV delay and its dependence on the type of outburst can be understood on the basis of qualitative considerations. Its quantitative discussion, however, requires model calculations and it is clear that the UV delays predicted from such models crucially depend on their assumptions and approximations. The most important of them are the outer boundary conditions. Of all models calculated in the past, only those by Smak (1984; also models discussed in the present paper), Ichikawa and Osaki (1992), and Hameury et al. (1998a, 1998b) used the correct outer boundary conditions, describing the deposition of the stream material in the outermost parts of the disk, and the effects of the angular momentum. Those effects involve: (a) the tidal removal of the angular momentum from the outer parts of the disk, (b) the specific angular momentum of the matter added by the stream being lower than that of the disk outer parts, and (c) the outside transfer of the angular momentum within the disk (which becomes dominant during outbursts). Their combined action not only causes the disk expan-

13 Vol sion during outburst and slow contraction during quiescence, but also profoundly affects the structure and evolution of the disk, particularly its outer parts. This can be easily shown in the case of Type A outbursts and for this purpose we choose our Model 13. Fig. 4. Loops in the? T e diagram during Type A outburst (Model 13) at two radial distances (R in cm). The shape of the loops in the? T e diagram (Fig. 4) strongly depends on the radius (i.e., the distance from the central star) and, generally, significantly differs from the "schematic" picture used commonly to illustrate the limit cycle mechanism. In the case of the inner parts of the disk the transition to the hot branch, which occurs with const. (as in "schematic" picture), is followed by a major increase in and the resulting increase of T e. All this is due to a combined effect of short time scales and of the geometry (small R) and in the case of the Type A outburst occurs at its advanced stage, close to maximum. In the case of the outer parts (where the instability first begins) the situation is different. Due to longer thermal time scale (see comments below), the transition to the hot branch occurs slowly enough for the effects of enhanced accretion to play their important role: the surface density decreases rapidly making the maximum temperature reached on the hot branch much lower than expected from the "schematic" picture. Very important 4 here is the tidal removal of the angular momentum which depends on T e (see Eqs. (26) and (32) in Smak 1984) and takes place as soon as the temperature begins to increase at the very onset of the outburst making decrease. Concerning the thermal time scale, it is worth noting that it is rather long in this case not only due to large R. It also depends strongly on the degree of deviation from thermal equilibrium. As can be seen in Fig. 4, due to decreasing, the line representing

14 690 A. A. transition to the hot branch passes very close to the thermal equilibrium curve and this implies that the effective thermal time scale must be rather long. The global evolution of the disk temperature is presented in Fig. 5. It shows that the temperature of the outer parts of the disk never exceeds 10 4 K and this has profound consequences: the ultraviolet flux cannot yet compete with the visual flux and the resulting UV delay must be relatively large. In addition we should recall that close to 10 4 K the difference between black body and Kurucz fluxes is much smaller than at higher temperatures (Section 2.2) and this results in a relatively small difference between the two sets of UV delays obtained for Type A outbursts (Section 2.3). Fig. 5. Temperature profiles during Type A outburst (Model 13). In order of increasing temperature the four curves correspond to: (1) onset of instability (marked with a dot); (2) log f(5500 Å) =?1:0; (3) log f(1200 Å) =?1:0; and (4) visual maximum. In the case of Type B outbursts the situation is more complex and it is impossible to describe it in a similarly simple way. At least two points, however, can be made. First, that with temperatures of the inner parts of the disk becoming much higher, the resulting differences between UV delays calculated with black body and Kurucz fluxes are much larger (Section 2.3). Secondly, that the outer boundary conditions, which affect the structure and evolution of the disk, must also be important in this case On the "Problem of UV Delay" Most models calculated in the past, including all those which produced small UV delays, used "simplified" outer boundary conditions: the radius of the disk was kept fixed and the effects of tidal removal of the angular momentum were not taken into account. The consequence of these assumptions for Type A outbursts was a much larger increase of the temperature of the outer parts of disk, well above 10 4 K (e.g., Fig. 4 of Meyer and Meyer-Hofmeister 1989, or Fig. 5 of Pringle et

15 Vol al. 1986). Consequently, the resulting UV delays had to be systematically smaller (particularly in the case of Kurucz fluxes; see discussion in Section 4.1). Once such simplified assumptions were adopted, it was also almost inevitable that in many models the stream material was deposited in the middle of the disk. At higher accretion rates, leading normally to Type A outbursts, this forced the onset of instability to occur at intermediate radii: R inst =R d 0:6 (Meyer and Meyer-Hofmeister 1989) or even R inst =R d 0:23 and 0.34 (Cannizzo and Kenyon 1987). Apart from other effects, such outbursts (which can hardly be called "Type A") could not possibly produce large UV delays (simply on account of the much shorter travel time; see discussion of this point by Cannizzo and Kenyon 1987). Another important factor is the proper description of the non-equilibrium phases (involving the energy equation) during transition to the hot state. Crude treatment of this phase is likely to produce more "vertical" transition to the hot branch with the resulting higher temperature on the hot branch. As remarked by Mineshige (1988), the negative conclusions by Pringle et al. (1986) and by Cannizzo and Kenyon (1987) were likely to be, at least in part, due to their over-simplified treatment of the energy equation. Meyer-Hofmeister (1987) was the first to note that the shape of the standard? T e relation is also important. For example, let us consider the separation between its two critical points. It is obvious that larger values of log AB must produce loops leading to higher temperatures on the hot branch. This effect can be seen when comparing two models of Pringle et al. (1986): the UV delay produced by their Model III (Fig. 5) with very large log AB was much smaller than that in Model VI (Fig. 8) with a smaller value of log AB. The important role played by the shape of the? T e relation can also be illustrated by considering the results of model calculations by Mineshige (1986, 1988) which, in spite of their "simplified" outer boundary conditions, produced large UV delays, consistent with observations. This was due to the "stagnation" phase, at T e 6000 K, obtained by Mineshige, and described by him precisely in terms of the relevant physical mechanisms. It appears that the origin of this "stagnation" phase can also be understood in more qualitative terms as being primarily due to the specific shape of the? T e relation. Let us first note that the relation used by Mineshige (1988, see also Mineshige and Osaki 1985, Fig. 2) differs from those used by other authors: it has a double-s shape with a middle stable branch and four critical points. Then note that the part of the loop in the? T e plane, representing transition to the hot state, passes very closely to (or even intersects with) the equilibrium curve at point B (Figs. 2 and 12 in Mineshige and Osaki 1985) where T e 6000 K. With situation being so close to equilibrium the effective thermal time scale must have been very long indeed resulting in the "stagnation" phase at this temperature level. Finally, what greatly contributed to the confusion around the "problem of UV

16 692 A. A. delay", was the careless attitude, shown by many authors, when comparing model predictions with observations. This is best illustrated by the case of SS Cyg, which became the favored, "negative" example, just because of its record large UV delay amounting to about 1 day. The first and only models with long orbital periods and large disk radii applicable to this case are those presented in Section 2. All earlier models were calculated with smaller radii. It therefore turns out that all "negative" conclusions concerning SS Cyg were based either on wishful thinking or on nonexisting models! Note added after referee s report. The referee of this paper raised the problem connected with the use of the "old" 1979 Kurucz models asking specifically how the results presented above would be modified by the use of the "new" 1993 models. To begin with, it should be emphasized that our main intention was to follow the common practice of other authors to use either the black body case or model atmospheres, the specific choice of the 1979 Kurucz models resulting simply from the desire to use the same models that were used in earlier papers. Besides, it was felt that, in the context of present applications, the difference between results based on any two sets of model atmospheres will certainly be small compared to the difference between model atmospheres and the much cruder black body approximation. Nevertheless it appears appropriate to clarify now this point in a quantitative way. Fig. 6. The ultraviolet-to-visual flux ratio R = f(1200 Å)=f(5500 Å) as a function of temperature for the black body case and for the two sets of Kurucz models. As pointed out in Section 2.2, in the context of present applications the difference between different "flux vs. T e " calibrations shows up in the ratio of fluxes at the two wavelengths: R = (1200 Å)=f (5500 Å), and its dependence on temperature. Fig. 6 shows three R(T e ) relations calculated for the three cases now considered: the black body and the two sets of Kurucz models. First of all, we should note large, systematic differences between R(T e ) resulting from

17 Vol Kurucz models and from the black body approximation. As summarized already in Section 2.2, at low temperatures R(Kurucz) is much lower than R(BB); they become equal at about 9000 K, and at higher temperatures R(Kurucz) is much larger; important consequences of those differences have been discussed in Section 4. Concerning the two sets of Kurucz models we note that differences between R(Kurucz79) and R(Kurucz93) are, as expected, almost negligibly small. Acknowledgements. The author wishes to thank the referee for his/her critical comments and to Dr. Beata Mazur and Dr. Andrzej Sołtan for their help with the 1993 Kurucz models. REFERENCES Cannizzo, J.K. 1993, "Accretion Disks in Compact Stellar Systems", Ed. J.C. Wheeler, Singapore: World Scientific, 6. Cannizzo, J.K., Wheeler, J.C., and Polidan, R.S. 1986, Astrophys. J., 301, 634. Cannizzo, J.K., and Kenyon, S.J. 1987, Astrophys. J., 320, 319. Hameury, J.-M., Menou, K., Dubus, G., and Lasota, J.-P. 1998a, MNRAS, 298, Hameury, J.-M., Lasota, J.-P., and Dubus, G. 1998b, MNRAS, in press. Hassall, B.J.M., Pringle, J.E., and Verbunt, F. 1986, MNRAS, 221, 169. Ichikawa, S., and Osaki, Y. 1992, Publ. Astr. Soc. Japan, 44, 15. Kurucz, R.L. 1979, Astrophys. J. Suppl. Ser., 40, 1. Mauche, C.W. 1996, "Astrophysics in the Extreme Ultraviolet", Eds. S. Bowyer and R.F. Malina, Dordrecht: Kluwer, 317. Meyer, F., and Meyer-Hofmeister, E. 1981, Astron. Astrophys., 104, L10. Meyer, F., and Meyer-Hofmeister, E. 1989, Astron. Astrophys., 221, 36. Meyer-Hofmeister, E. 1987, Astron. Astrophys., 175, 113. Mineshige, S. 1986, "Hydrodynamic and Magnetohydrodynamic Problems in the Sun and Stars", Ed. Y. Osaki, Tokyo: Univ. of Tokyo, 275. Mineshige, S. 1988, Astron. Astrophys., 190, 72. Mineshige, S., and Osaki, Y. 1985, Publ. Astr. Soc. Japan, 37, 1. Osaki, Y. 1996, P.A.S.P., 108, 39. Polidan, R.S., and Holberg, J.B. 1987, MNRAS, 225, 131. Popper, D.M. 1980, Ann. Rev. Astron. Astrophys., 18, 115. Pringle, J.E., Verbunt, F., and Wade, R.A. 1986, MNRAS, 221, 169. Schwarzenberg-Czerny, A., Ward, M., Hanes, D.A., Jones, D.H.P., Pringle, J.E., Verbunt, F., and Wade, R.A. 1985, MNRAS, 212, 645. Sion, E.M. 1991, Astron. J., 102, 295. Smak, J. 1984, Acta Astron., 34, 161. Smak, J. 1987, Astrophys. Space Sci., 131, 497. Smak, J. 1992, Acta Astron., 42, 323. Verbunt, F., et al. 1984, MNRAS, 210, 197. Wheatley, P.J., Verbunt, F., Belloni, T., Watson, M.G., Naylor, T., Ishida, M., Duck, S.R., and Pfeffermann, E. 1996, Astron. Astrophys., 307, 137.