- Email: [email protected]
Unsolved problems of dwarf nova outbursts
New Astronomy Reviews 44 (2000) 171–175 www.elsevier.nl / locate / newar
Unsolved problems of dwarf nova outbursts J. Smak 1 Copernicus Astronomical Center, Bartycka 18, 00 -716 Warsaw, Poland
Abstract Four problems are discussed. (1) Model light curves show significant increase of the disk luminosity during quiescence, an effect which is not present in the observed light curves. It is suggested that the slope of the lower branch of the S 2 T e relation should be significantly decreased. (2) The width – Porb relation for narrow outbursts is well reproduced with model data for ahot 5 0.2. The bimodal distribution of outburst durations and, in particular, the origin of wide outbursts and the nature of their width – Porb relation, require explanation. (3) It is suggested that problems with the thermal-tidal instability (TTI) model for superoutbursts might be solved by a hybrid model, combining the TTI model with the irradiation-enhanced mass-transfer model. A strong argument in favour of irradiation is provided by the ratio of the irradiating flux to the intrinsic flux of the secondary component, which turns out to be very large in the case of dwarf novae showing superoutbursts, with U Gem being a borderline case. (4) Characteristic time-scales observed during dwarf nova outbursts depend on the viscous time-scale, allowing an empirical determination of a. Three independent determinations, based on the rates of decline following outburst maximum, the UV delay observed during rising light, and the widths of outbursts, give consistently ahot ¯ 0.2. It should be added, however, that those time-scales depend also strongly on the radius of the disk. In this context it is disturbing to note that the observed disk radii appear to be smaller than those resulting from model calculations. 2000 Elsevier Science B.V. All rights reserved.
1. Luminosity of the disk at quiescence All model light curves (for references to individual papers see Cannizzo (1993) and Osaki (1996)) show that the luminosity of the disk increases during quiescence, in some cases by as much as 1–2 magnitudes. This is, obviously, a direct consequence of the slope of the lower branch of the S 2 T e relation: as the surface density increases anywhere in the disk, so does the effective temperature and this shows up in the global luminosity of the disk. Before comparing those predictions with the observed light curves one should account for the contributions from the two stellar components and the hot spot. In general, at longer orbital periods it is 1
E-mail: [email protected]
the secondary which is the main contributor in the visual part of the spectrum, often providing as much as 50 percent of the total flux and at the shortest orbital periods the same is true for the primary component. An inspection of relevant data for individual systems shows that the disk contributes no less than 1 / 2–1 / 3 of the total flux. Using these estimates we obtain that the effect of the increasing disk luminosity should show up in the observed light curves at a level of about 0.5 mag. and therefore should be easily detectable. All well covered light curves of dwarf novae show, however, that the light at quiescence is constant to within 0.1 mag. There is only one way of removing this discrepancy: the cool branch of the S 2 T e relation must be made considerably flatter. Its slope should be determined from a detailed analysis of data available for systems with well determined parame-
1387-6473 / 00 / $ – see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S1387-6473( 00 )00033-6
172
J. Smak / New Astronomy Reviews 44 (2000) 171 – 175
ters. In addition, its ‘mean’ effective temperature should be about 4000–50008 K, as required by direct determinations of the disk temperature at quiescence (Wood et al., 1986; Wood et al., 1989). Such a correction, while removing the existing discrepancy, will also be another step in the empirical determination of the viscosity (or, at least, its dependence on temperature). One should recall that the first such step had already been made in the early 1980s, when it was found that the S 2 T e relation based on a single value of a is insufficient to reproduce the dwarf nova outbursts. Consequently, a modification of the shape of the S 2 T e relation was then adopted, commonly described in terms of two different values of a for the hot and cool branch, with ahot ¯ 4 2 5 acool . It should be noted at this point that, with further modification proposed here, it will no longer be possible to describe the shape of the cool branch of the S 2 T e relation with acool 5 const.
2. Narrow and wide outbursts One of the characteristic features of dwarf nova light curves is that they are not strictly periodic: for a given system, the length of the cycle and the shape of the outburst curve differ considerably from one cycle to another. In many cases the distribution of outburst durations (or widths) is clearly bimodal (van Paradijs, 1983; Szkody & Mattei, 1984), with the amplitudes of the ‘narrow’ and ‘wide’ outbursts differing by only 0.2 mag. It is important to note that such a bimodal behaviour is shown by systems with Type A (outside-in) outbursts (e.g. U Gem), as well as those with Type B (inside-out) outbursts (e.g. SS Cyg). Model light curves (again, for references to individual papers, see Cannizzo (1993) and Osaki (1996)) do not reproduce this phenomenon. To begin with, in the case of Type A outbursts the light curves are nearly strictly periodic, with consecutive outbursts being of identical shape. In the case of Type B outbursts the situation seems to be better: light curves show ‘alternating’ outbursts of different shapes, with the short (or narrow) outbursts corresponding usually to a situation when the instability wave does not propagate all the way to the outer
parts of the disk. However, the amplitudes of those narrow outbursts are considerably smaller (often by several magnitudes) than those of wide outbursts. There is one important factor not included in the ‘standard’ dwarf nova models, namely the irradiation-induced enhancement of the mass transfer from the secondary. To check the importance of this effect, several numerical experiments have been ~ tr performed, with an arbitrary enhancement of M during outbursts. The input parameters were chosen to produce a ‘standard’ model showing alternating Type B outbursts of different durations and amplitudes. With a moderate (factor of 2) enhancement the outburst amplitudes became identical (as desired), but so did the outburst durations, their widths being practically the same as those of ‘wider’ outbursts produced by the original, ‘standard’ model. Closer analysis shows that the main effect of the moderately ~ tr is to help the instability wave propenhanced M agate all the way to the outer edge of the disk and, in particular, that the outburst durations simply correspond to the disk’s viscous time-scale. Observational ~ tr show that such an enhancement does data on M indeed take place (Smak, 1996a). Therefore, it appears that the low amplitude, very narrow outbursts, produced by the ‘standard’ model, are only artefacts resulting from its incompletness. Consequently, we must conclude that what should be compared with observations in the case of Type B outbursts are either the durations produced by a ~ tr , or – model including a moderately enhanced M since the two are practically identical – the durations of ‘wider’ outbursts produced by the ‘standard’ model. This will be done below. It should be added, that any additional effect, such as a major, prolonged ~ tr (see below), can only make an enhancement of M outburst last longer. ~ tr Further numerical experiments with enhanced M show that the situation becomes qualitatively different when this enhancement is large enough to result in a steady-state accretion. Then the outburst amplitude becomes slightly larger and – more importantly – its duration, instead of being controlled by the disk itself, must be controlled by the secondary component: the outburst can end only when ~ tr returns to a much lower level. Note that such a M scenario would closely resemble the model proposed
J. Smak / New Astronomy Reviews 44 (2000) 171 – 175
by Meyer & Meyer-Hofmeister (1983) for the Z Cam behaviour. However, how it could explain the bimodal distribution of outburst widths – is an open question. The durations (or widths) of narrow and wide outbursts are strongly correlated with the orbital period. This is shown in Fig. 1, with observational data taken from Van Paradijs (1983, Table 1). Its left panel shows a comparison with model data (Smak, 1998) calculated with ahot 5 0.2 (the outbursts widths, to be comparable to the observed values, were determined from model light curves at a level 2 mag. below maximum). The agreement is almost perfect and shows that the width of narrow outbursts can be reproduced with models using the value of ahot which is consistent with other estimates (see Section 4). The problem then is how to explain the longer duration of wide outbursts. Van Paradijs (1983) suggested that wide outbursts, observed in systems with longer orbital periods, could be essentially the same phenomenon as superoutbursts observed in ultra-short-period SU
173
UMa stars. We now have an additional evidence for TU Men and U Gem, which shows that this is not the case: TU Men shows three types of outbursts: narrow, wide and superoutbursts (Bateson, 1979; Bateson, 1991), while U Gem showed its famous superoutburst in 1985, which lasted for about 40 days (Mason et al., 1988; Kuulkers, 1999). Consequently, as the right panel of Fig. 1 shows, there appear to be two separate width – Porb relations for wide outbursts and for superoutbursts.
3. The superoutbursts The thermal-tidal instability (TTI) model (Osaki, 1989; Osaki, 1996) combines the thermal instability of dwarf nova disks with their tidal instability, which occurs – due to the 3:1 resonance – in large disks in systems with mass-ratios q , qcrit ¯ 0.25. As discussed earlier (Smak, 1996a), this model faces a serious problem with the sequence of events it predicts: the superhumps should appear at an early
Fig. 1. The widths of outbursts are plotted against the orbital period. Left: Narrow outbursts. Observational data are shown as crosses. Model data, with ahot 5 0.2, are shown as filled squares and triangles corresponding, respectively, to Type A and Type B outbursts. Right: Wide outbursts (crosses) and superoutbursts (asterisks). The asterisk with an arrow represents the 1985 superoutburst of U Gem which lasted for about 40 days. The dotted line is the correlation defined by narrow outbursts.
174
J. Smak / New Astronomy Reviews 44 (2000) 171 – 175
phase of a superoutburst (certainly not later than its maximum), while observations show that this happens only one or two days after maximum. An equally serious problem is connected with superoutbursts of two dwarf novae mentioned already in Section 2: TU Men and U Gem. The mass-ratio of U Gem is q 5 0.46. For TU Men no definite determinations of system parameters are available, but it appears that its mass-ratio must be within the range q 5 0.33–0.46 (Mennickent, 1995). Hence in both cases the mass-ratio is larger than the critical value essential for the TTI model to work. It has been suggested (Smak, 1996a) that a solution to these problems could be provided by a hybrid model, combining the tidal instability mechanism of the TTI model with the enhanced masstransfer model, proposed earlier also by Osaki (1985). The importance of irradiation of the secondary component in different systems can be measured by the ratio of the mean irradiating flux kFirr l at the peak of an outburst to the intrinsic stellar flux F2 . This ratio was calculated for several dwarf novae with reliable system parameters taken from the catalogue by Ritter & Kolb (1998); TU Men was also included using a probable range of its parameters. The results, shown in Fig. 2, are highly suggestive: In the case of dwarf novae with superout-
bursts the ratio kFirr l /F2 is indeed much larger than in other cases. And U Gem, with its single superoutburst, appears to be a borderline case!
4. The time-scales and the value of a Characteristic time-scales observed during dwarf nova outbursts depend on the viscous time-scale, defined by the parameter ahot on the hot branch of the S 2 T e relation. This provides an important opportunity for an empirical determination of a. All such determinations give consistent results: ahot ¯ 0.2 from the rates of decline following outburst maximum (Smak, 1984), ahot 5 0.1–0.2 from the UV delay observed during rising light (Smak, 1998), and ahot ¯ 0.2 from the widths of outbursts, as shown in Fig. 1 and discussed in Section 2. (To avoid possible confusion with various practical definitions of a (Osaki, 1996), it is necessary to add that the values quoted above correspond to a definition of a via the viscous stress: w 5 tr, f 5 a P.) It should be kept in mind, however, that the observed time-scales depend also on the radius of the disk, this dependence being responsible for the observed correlations with the orbital period. The nature of this dependence can be illustrated for the case of Type A outbursts, where one can assume that the outburst durations or UV delays are related to the travel time of the accretion wave from R d to the inner radius R in . This travel time, estimated analytically (Smak, 1998), is: 3/2 Dt (R d ,R in ) | a 20.7 (R d3 / 2 2 R in ) ¯ a 20.7 R d3 / 2 .
Fig. 2. The ratio kFirr l /F2 of the mean irradiating flux at the peak of an outburst to the intrinsic stellar flux is plotted against the orbital period. Filled squares are systems with superoutbursts. Open squares – other dwarf novae.
(1)
With such a strong dependence on R d , it is obvious that dwarf nova models used to obtain estimates of a should include proper treatment of the radius of the disk and its variations during the outburst cycle. This condition is satisfied only by models which use the correct outer boundary conditions (Smak, 1984; Smak, 1998; Ichikawa & Osaki, 1992; Hameury et al., 1998; Hameury et al., 1999), involving the following effects: (a) the tidal removal of the angular momentum from the outer parts of the disk, (b) the deposition of the stream material, with its specific angular momentum being
J. Smak / New Astronomy Reviews 44 (2000) 171 – 175
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 is to cause the disk to expand during outburst and slowly contract during quiescence, and such variations are indeed observed in dwarf novae (cf. Smak, 1996b). Tidal effects were discussed analytically by Papaloizou & Pringle (1977), their results showing that the effective tidal radius is close to the Roche lobe: R tid | 0.9R Roche (where R Roche is the mean radius of the Roche lobe), coinciding with the largest non-intersecting orbits in the three-body problem ´ (e.g. Paczy nski, 1977; Smak, 1976). Dwarf nova models, using an approximate formula (Smak, 1984) based on these results, produce disk radius variations between R min ¯ 0.8R Roche and R max ¯ 0.95R Roche . Observational determinations of disk radii are available for eclipsing cataclysmic variables. Recent compilation of all such determinations and estimates (Harrop-Allin & Warner, 1996) shows that, generally, disks appear to be smaller than theoretical predictions. In the case of the best documented dwarf novae, the radii of their disks are found to vary during outburst cycle between R min ¯ 0.5 2 0.7R Roche and R max ¯ 0.9R Roche (Smak, 1996b). These discrepancies and their origin deserve further, more detailed analysis. There are several possibilities to be considered. One of them could be that our ‘prescription’ for the tidal effects should be revised. In the case of dwarf novae, however, we should note that during outbursts, when the disk is large and therefore the tidal effects are very important, the observed values of R max appear to be roughly consistent with theoretical predictions. It is only during quiescence that the observed and modelpredicted values of R min show significant disagreement. At this point it should be recalled that observational determinations of disk radii in dwarf novae are based on eclipses of the hot spot, with an implicit assumption that the ‘distance’ of the spot from the
175
white dwarf is representative for the radius of the disk. This may not necessarily be true, particularly during quiescence, when the disk is geometrically thin and of low density, and when the hot spot may not necessarily be located exactly at the outer edge of the disk.
References Bateson, F.M., 1979, Pub. Var. Star Section, RAS New Zealand, 7, 5. Bateson, F.M., 1991, Pub. Var. Star Section, RAS New Zealand, 16, 40. Cannizzo, J.K., 1993, in: J.C.Wheeler, (Ed.), Accretion Disks in Compact Stellar Systems, World Scientific, Singapore, p. 6. ´ Hameury, J.-M., Menou, K., Dubus, G., Lasota, J.-P., & Hure, J.-M., 1998, MNRAS, 298, 1048. Hameury, J.-M., Lasota, J.-P., & Dubus, G., 1999, MNRAS, 303, 39. Harrop-Allin, M.K. & Warner, B., 1996, MNRAS, 279, 219. Ichikawa, S. & Osaki, Y., 1992, PASJ, 44, 15. Kuulkers, E., 1999, in: S. Mineshige & J.C. Wheeler, (Eds.), Disk Instabilities in Close Binary Systems, Universal Academy Press, Tokyo, p. 169. Mason, K.O., Cordova, F.A., Watson, M.G., & King, A.R., 1988, MNRAS, 232, 779. Mennickent, R.E., 1995, A&A, 294, 126. Meyer, F. & Meyer-Hofmeister, E., 1983, A&A, 121, 29. Osaki, Y., 1985, A&A, 144, 369. Osaki, Y., 1989, PASJ, 41, 1005. Osaki, Y., 1996, PASP, 108, 39. ´ Paczynski, B., 1977, ApJ, 216, 822. Papaloizou, J. & Pringle, J.E., 1977, MNRAS, 181, 441. Ritter, H. & Kolb, U., 1998, A&A, Suppl. Ser., 129, 83. Smak, J., 1976, AcA, 26, 277. Smak, J., 1984, AcA, 34, 161. Smak, J., 1996a, in: A. Evans & J.H. Wood, (Eds.), Cataclysmic Variables and Related Objects, Kluwer, Dordrecht, p. 45. Smak, J., 1996b, AcA, 46, 377. Smak, J., 1998, AcA, 48, 677. Szkody, P. & Mattei, J.A., 1984, PASP, 96, 988. van Paradijs, J., 1983, A&A, 125, L16. Wood, J.A., Horne, K., Berriman, G., Wade, R., O’Donoghue, D., & Warner, B., 1986, MNRAS, 219, 629. Wood, J.A., Horne, K., Berriman, G., & Wade, R., 1989, ApJ, 341, 974.
Unsolved problems of dwarf nova outbursts J. Smak 1 Copernicus Astronomical Center, Bartycka 18, 00 -716 Warsaw, Poland
Abstract Four problems are discussed. (1) Model light curves show significant increase of the disk luminosity during quiescence, an effect which is not present in the observed light curves. It is suggested that the slope of the lower branch of the S 2 T e relation should be significantly decreased. (2) The width – Porb relation for narrow outbursts is well reproduced with model data for ahot 5 0.2. The bimodal distribution of outburst durations and, in particular, the origin of wide outbursts and the nature of their width – Porb relation, require explanation. (3) It is suggested that problems with the thermal-tidal instability (TTI) model for superoutbursts might be solved by a hybrid model, combining the TTI model with the irradiation-enhanced mass-transfer model. A strong argument in favour of irradiation is provided by the ratio of the irradiating flux to the intrinsic flux of the secondary component, which turns out to be very large in the case of dwarf novae showing superoutbursts, with U Gem being a borderline case. (4) Characteristic time-scales observed during dwarf nova outbursts depend on the viscous time-scale, allowing an empirical determination of a. Three independent determinations, based on the rates of decline following outburst maximum, the UV delay observed during rising light, and the widths of outbursts, give consistently ahot ¯ 0.2. It should be added, however, that those time-scales depend also strongly on the radius of the disk. In this context it is disturbing to note that the observed disk radii appear to be smaller than those resulting from model calculations. 2000 Elsevier Science B.V. All rights reserved.
1. Luminosity of the disk at quiescence All model light curves (for references to individual papers see Cannizzo (1993) and Osaki (1996)) show that the luminosity of the disk increases during quiescence, in some cases by as much as 1–2 magnitudes. This is, obviously, a direct consequence of the slope of the lower branch of the S 2 T e relation: as the surface density increases anywhere in the disk, so does the effective temperature and this shows up in the global luminosity of the disk. Before comparing those predictions with the observed light curves one should account for the contributions from the two stellar components and the hot spot. In general, at longer orbital periods it is 1
E-mail: [email protected]
the secondary which is the main contributor in the visual part of the spectrum, often providing as much as 50 percent of the total flux and at the shortest orbital periods the same is true for the primary component. An inspection of relevant data for individual systems shows that the disk contributes no less than 1 / 2–1 / 3 of the total flux. Using these estimates we obtain that the effect of the increasing disk luminosity should show up in the observed light curves at a level of about 0.5 mag. and therefore should be easily detectable. All well covered light curves of dwarf novae show, however, that the light at quiescence is constant to within 0.1 mag. There is only one way of removing this discrepancy: the cool branch of the S 2 T e relation must be made considerably flatter. Its slope should be determined from a detailed analysis of data available for systems with well determined parame-
1387-6473 / 00 / $ – see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S1387-6473( 00 )00033-6
172
J. Smak / New Astronomy Reviews 44 (2000) 171 – 175
ters. In addition, its ‘mean’ effective temperature should be about 4000–50008 K, as required by direct determinations of the disk temperature at quiescence (Wood et al., 1986; Wood et al., 1989). Such a correction, while removing the existing discrepancy, will also be another step in the empirical determination of the viscosity (or, at least, its dependence on temperature). One should recall that the first such step had already been made in the early 1980s, when it was found that the S 2 T e relation based on a single value of a is insufficient to reproduce the dwarf nova outbursts. Consequently, a modification of the shape of the S 2 T e relation was then adopted, commonly described in terms of two different values of a for the hot and cool branch, with ahot ¯ 4 2 5 acool . It should be noted at this point that, with further modification proposed here, it will no longer be possible to describe the shape of the cool branch of the S 2 T e relation with acool 5 const.
2. Narrow and wide outbursts One of the characteristic features of dwarf nova light curves is that they are not strictly periodic: for a given system, the length of the cycle and the shape of the outburst curve differ considerably from one cycle to another. In many cases the distribution of outburst durations (or widths) is clearly bimodal (van Paradijs, 1983; Szkody & Mattei, 1984), with the amplitudes of the ‘narrow’ and ‘wide’ outbursts differing by only 0.2 mag. It is important to note that such a bimodal behaviour is shown by systems with Type A (outside-in) outbursts (e.g. U Gem), as well as those with Type B (inside-out) outbursts (e.g. SS Cyg). Model light curves (again, for references to individual papers, see Cannizzo (1993) and Osaki (1996)) do not reproduce this phenomenon. To begin with, in the case of Type A outbursts the light curves are nearly strictly periodic, with consecutive outbursts being of identical shape. In the case of Type B outbursts the situation seems to be better: light curves show ‘alternating’ outbursts of different shapes, with the short (or narrow) outbursts corresponding usually to a situation when the instability wave does not propagate all the way to the outer
parts of the disk. However, the amplitudes of those narrow outbursts are considerably smaller (often by several magnitudes) than those of wide outbursts. There is one important factor not included in the ‘standard’ dwarf nova models, namely the irradiation-induced enhancement of the mass transfer from the secondary. To check the importance of this effect, several numerical experiments have been ~ tr performed, with an arbitrary enhancement of M during outbursts. The input parameters were chosen to produce a ‘standard’ model showing alternating Type B outbursts of different durations and amplitudes. With a moderate (factor of 2) enhancement the outburst amplitudes became identical (as desired), but so did the outburst durations, their widths being practically the same as those of ‘wider’ outbursts produced by the original, ‘standard’ model. Closer analysis shows that the main effect of the moderately ~ tr is to help the instability wave propenhanced M agate all the way to the outer edge of the disk and, in particular, that the outburst durations simply correspond to the disk’s viscous time-scale. Observational ~ tr show that such an enhancement does data on M indeed take place (Smak, 1996a). Therefore, it appears that the low amplitude, very narrow outbursts, produced by the ‘standard’ model, are only artefacts resulting from its incompletness. Consequently, we must conclude that what should be compared with observations in the case of Type B outbursts are either the durations produced by a ~ tr , or – model including a moderately enhanced M since the two are practically identical – the durations of ‘wider’ outbursts produced by the ‘standard’ model. This will be done below. It should be added, that any additional effect, such as a major, prolonged ~ tr (see below), can only make an enhancement of M outburst last longer. ~ tr Further numerical experiments with enhanced M show that the situation becomes qualitatively different when this enhancement is large enough to result in a steady-state accretion. Then the outburst amplitude becomes slightly larger and – more importantly – its duration, instead of being controlled by the disk itself, must be controlled by the secondary component: the outburst can end only when ~ tr returns to a much lower level. Note that such a M scenario would closely resemble the model proposed
J. Smak / New Astronomy Reviews 44 (2000) 171 – 175
by Meyer & Meyer-Hofmeister (1983) for the Z Cam behaviour. However, how it could explain the bimodal distribution of outburst widths – is an open question. The durations (or widths) of narrow and wide outbursts are strongly correlated with the orbital period. This is shown in Fig. 1, with observational data taken from Van Paradijs (1983, Table 1). Its left panel shows a comparison with model data (Smak, 1998) calculated with ahot 5 0.2 (the outbursts widths, to be comparable to the observed values, were determined from model light curves at a level 2 mag. below maximum). The agreement is almost perfect and shows that the width of narrow outbursts can be reproduced with models using the value of ahot which is consistent with other estimates (see Section 4). The problem then is how to explain the longer duration of wide outbursts. Van Paradijs (1983) suggested that wide outbursts, observed in systems with longer orbital periods, could be essentially the same phenomenon as superoutbursts observed in ultra-short-period SU
173
UMa stars. We now have an additional evidence for TU Men and U Gem, which shows that this is not the case: TU Men shows three types of outbursts: narrow, wide and superoutbursts (Bateson, 1979; Bateson, 1991), while U Gem showed its famous superoutburst in 1985, which lasted for about 40 days (Mason et al., 1988; Kuulkers, 1999). Consequently, as the right panel of Fig. 1 shows, there appear to be two separate width – Porb relations for wide outbursts and for superoutbursts.
3. The superoutbursts The thermal-tidal instability (TTI) model (Osaki, 1989; Osaki, 1996) combines the thermal instability of dwarf nova disks with their tidal instability, which occurs – due to the 3:1 resonance – in large disks in systems with mass-ratios q , qcrit ¯ 0.25. As discussed earlier (Smak, 1996a), this model faces a serious problem with the sequence of events it predicts: the superhumps should appear at an early
Fig. 1. The widths of outbursts are plotted against the orbital period. Left: Narrow outbursts. Observational data are shown as crosses. Model data, with ahot 5 0.2, are shown as filled squares and triangles corresponding, respectively, to Type A and Type B outbursts. Right: Wide outbursts (crosses) and superoutbursts (asterisks). The asterisk with an arrow represents the 1985 superoutburst of U Gem which lasted for about 40 days. The dotted line is the correlation defined by narrow outbursts.
174
J. Smak / New Astronomy Reviews 44 (2000) 171 – 175
phase of a superoutburst (certainly not later than its maximum), while observations show that this happens only one or two days after maximum. An equally serious problem is connected with superoutbursts of two dwarf novae mentioned already in Section 2: TU Men and U Gem. The mass-ratio of U Gem is q 5 0.46. For TU Men no definite determinations of system parameters are available, but it appears that its mass-ratio must be within the range q 5 0.33–0.46 (Mennickent, 1995). Hence in both cases the mass-ratio is larger than the critical value essential for the TTI model to work. It has been suggested (Smak, 1996a) that a solution to these problems could be provided by a hybrid model, combining the tidal instability mechanism of the TTI model with the enhanced masstransfer model, proposed earlier also by Osaki (1985). The importance of irradiation of the secondary component in different systems can be measured by the ratio of the mean irradiating flux kFirr l at the peak of an outburst to the intrinsic stellar flux F2 . This ratio was calculated for several dwarf novae with reliable system parameters taken from the catalogue by Ritter & Kolb (1998); TU Men was also included using a probable range of its parameters. The results, shown in Fig. 2, are highly suggestive: In the case of dwarf novae with superout-
bursts the ratio kFirr l /F2 is indeed much larger than in other cases. And U Gem, with its single superoutburst, appears to be a borderline case!
4. The time-scales and the value of a Characteristic time-scales observed during dwarf nova outbursts depend on the viscous time-scale, defined by the parameter ahot on the hot branch of the S 2 T e relation. This provides an important opportunity for an empirical determination of a. All such determinations give consistent results: ahot ¯ 0.2 from the rates of decline following outburst maximum (Smak, 1984), ahot 5 0.1–0.2 from the UV delay observed during rising light (Smak, 1998), and ahot ¯ 0.2 from the widths of outbursts, as shown in Fig. 1 and discussed in Section 2. (To avoid possible confusion with various practical definitions of a (Osaki, 1996), it is necessary to add that the values quoted above correspond to a definition of a via the viscous stress: w 5 tr, f 5 a P.) It should be kept in mind, however, that the observed time-scales depend also on the radius of the disk, this dependence being responsible for the observed correlations with the orbital period. The nature of this dependence can be illustrated for the case of Type A outbursts, where one can assume that the outburst durations or UV delays are related to the travel time of the accretion wave from R d to the inner radius R in . This travel time, estimated analytically (Smak, 1998), is: 3/2 Dt (R d ,R in ) | a 20.7 (R d3 / 2 2 R in ) ¯ a 20.7 R d3 / 2 .
Fig. 2. The ratio kFirr l /F2 of the mean irradiating flux at the peak of an outburst to the intrinsic stellar flux is plotted against the orbital period. Filled squares are systems with superoutbursts. Open squares – other dwarf novae.
(1)
With such a strong dependence on R d , it is obvious that dwarf nova models used to obtain estimates of a should include proper treatment of the radius of the disk and its variations during the outburst cycle. This condition is satisfied only by models which use the correct outer boundary conditions (Smak, 1984; Smak, 1998; Ichikawa & Osaki, 1992; Hameury et al., 1998; Hameury et al., 1999), involving the following effects: (a) the tidal removal of the angular momentum from the outer parts of the disk, (b) the deposition of the stream material, with its specific angular momentum being
J. Smak / New Astronomy Reviews 44 (2000) 171 – 175
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 is to cause the disk to expand during outburst and slowly contract during quiescence, and such variations are indeed observed in dwarf novae (cf. Smak, 1996b). Tidal effects were discussed analytically by Papaloizou & Pringle (1977), their results showing that the effective tidal radius is close to the Roche lobe: R tid | 0.9R Roche (where R Roche is the mean radius of the Roche lobe), coinciding with the largest non-intersecting orbits in the three-body problem ´ (e.g. Paczy nski, 1977; Smak, 1976). Dwarf nova models, using an approximate formula (Smak, 1984) based on these results, produce disk radius variations between R min ¯ 0.8R Roche and R max ¯ 0.95R Roche . Observational determinations of disk radii are available for eclipsing cataclysmic variables. Recent compilation of all such determinations and estimates (Harrop-Allin & Warner, 1996) shows that, generally, disks appear to be smaller than theoretical predictions. In the case of the best documented dwarf novae, the radii of their disks are found to vary during outburst cycle between R min ¯ 0.5 2 0.7R Roche and R max ¯ 0.9R Roche (Smak, 1996b). These discrepancies and their origin deserve further, more detailed analysis. There are several possibilities to be considered. One of them could be that our ‘prescription’ for the tidal effects should be revised. In the case of dwarf novae, however, we should note that during outbursts, when the disk is large and therefore the tidal effects are very important, the observed values of R max appear to be roughly consistent with theoretical predictions. It is only during quiescence that the observed and modelpredicted values of R min show significant disagreement. At this point it should be recalled that observational determinations of disk radii in dwarf novae are based on eclipses of the hot spot, with an implicit assumption that the ‘distance’ of the spot from the
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white dwarf is representative for the radius of the disk. This may not necessarily be true, particularly during quiescence, when the disk is geometrically thin and of low density, and when the hot spot may not necessarily be located exactly at the outer edge of the disk.
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