Dwarf Nova Outbursts. III. The Viscosity Parameter. J. S m a k

ACTA ASTRONOMICA Vol. 49 (1999) pp. 391 401 Dwarf Nova Outbursts. III. The Viscosity Parameter by J. S m a k N. Copernicus Astronomical Center, Polish Academy of Sciences, ul. Bartycka 18, 00-716 Warsaw, Poland e-mail: jis@camk.edu.pl Received July 9, 1999 ABSTRACT Visual light curves based on dwarf nova models, covering a broad range of relevant parameters, are used to determine the widths of outbursts, rates of decline, and UV delays, as well as their dependence on the orbital period. From a comparison with the observational data it is found that the value of on the hot branch of the? T e relation is hot 0:2. An important test is also made concerning the tidal radii of disks. A comparison of the absolute visual magnitudes at maximum, obtained from model light curves, with the observed M V (max) gives R tid=r Roche 0:9, which is the standard value used in model calculations. Key words: Accretion, accretion disks binaries: close novae, cataclysmic variables 1. Introduction Characteristic time-scales observed during dwarf nova outbursts depend on the viscous time-scale, defined by the parameter hot on the hot branch of the? T e relation. This provides an important and unique opportunity of an empirical determination of. This will be done in the present paper by discussing the widths (or durations) of outbursts, the rates of decline following outburst maximum, and the UV delays observed during rising light, and their correlations with the orbital period. In particular, the values of hot will be determined by comparing the observed correlations with their theoretical counterparts, resulting from model calculations. 2. Analytical Considerations As discussed in Paper I (Smak 1998), the characteristic, effective time scales observed during Type A outbursts, can be related to the travel time of the accretion

392 A. A. wave. Type B outbursts are more complicated but, even in their case, an estimate of the travel time should give at least some information concerning the dependence on the relevant parameters. Using the -disk approximation we estimate the travel time as follows. The radial velocity of the accreting matter is GM1 1=2 zo 2 V r (1) R R where = hot, corresponds to the hot branch of the? T e relation, which is relevant during outburst. z o =R can be eliminated from Eq. (1) using its dependence on other parameters, as obtained from numerical integrations of the vertical structure of the disk (Smak 1992). As compared to Paper I, where only was considered, we now take into account the full dependence on T e, R, M 1, and. Eliminating T e with the standard expression for stationary accretion 4 T e M1 MR _?3 1? (R 1 =R) 1=2 ; (2) and using an approximate treatment of the square bracket in Eq. (2), we get V r 0:7 M 1 0:46 _M 0:37 R?0:33 : (3) Integrating this equation we obtain for the travel time between R d and the inner radius R in t(r d ; R in )?0:7 _ M?0:37 M 1?0:46 (Rd 1:33? Rin 1:33 ): (4) In addition to the explicit dependence on M 1 and M _, the following differences, as compared with Eq. (2) of Paper I, should be noted. First of all, there was a misprint in Eq. (2) of Paper I: 0:7 instead of?0:7. Secondly, the exponent at R in Eq. (4) is slightly smaller (1.33 instead of 1.5); this is due to the present taking into account the dependence of z o =R on R. For the purpose of further discussion we now neglect R in in Eq. (4). Furthermore we assume that R d = R tid, where R tid is the tidal radius, assumed to be a certain fraction of the effective radius of the Roche lobe around the primary component. Using the Kepler Law we can write R tid R Roche = Af (q) f (q)(1 + q) 1=3 M 1 1=3 P orb 2=3 (5) where f (q) = R Roche =A is the dimensionless radius of the Roche lobe. Crude estimates show that f (q)(1 + q) 1=3 depends only weakly ( P orb?0:11 ) on the orbital period. Substituting this dependence into Eq. (5) and inserting the resulting R tid into Eq. (4) we get t?0:7 _M?0:37 M 1?0:02 P orb 0:74 : (6)

Vol. 49 393 This gives the predicted form of the dependence of the observed time scales on the orbital period, the viscosity parameter = hot, and the accretion rate M _. Note that there is practically no dependence on the mass of the primary component. At this point it would be natural to try to eliminate _M from Eq. (6). In the simplest approach, we could try to estimate it, and its possible dependence on P orb, from M V (max) (see, for example, Fig. 1 in Warner 1987). One should note, however, that M _, appearing in Eq. (6), does not really correspond to M _ (max) = f [M V (max)]. In fact, its physical meaning is not as obvious as it may seem. Going back to Eq. (1) we recall that what was needed there was z o =R, corresponding to the local structure of the transition front as it moves across the disk. In our approach it was eliminated via the adopted dependence of z o =R on T e, and T e on M _ (and this is how M _ appeared in Eq. 6). This was only a crude approximation but, unfortunately, there seems to be no other (better) way of estimating z o =R analytically. One could, of course, use here the results of model calculations but, if so, our considerations would no longer be purely analytical. 3. Models Models which will be used in further Sections for comparison with observational data are the same as those described in Paper I. It can be added here that the masses of the primary component used in those calculations, namely M 1 = 0:6M at P orb = 0:057 d and M 1 = 1:0M at P orb > 0:1 d, are consistent with observational evidence available for dwarf novae of different orbital periods (e.g., Webbink 1990, Warner 1995, Ritter and Kolb 1998). T a b l e 1 Model Data P hr mod W 2 mod W 2 1.38 11 2.34 0.73 21 1.36 0.41 1.38 12 3.09 0.65 22 1.92 0.38 4.20 13 8.90 3.15 23 5.12 1.81 4.20 14 13.31 2.94 24 7.26 1.64 5.82 16 15.46 3.62 26 8.74 2.03 7.50 18 19.83 6.46 28 11.15 3.56 7.50 18d 19.51 5.70 28d 10.87 3.17 Visual system light curves, calculated from those models using fluxes from Kurucz (1993) model atmospheres, were now used to determine (1) the widths of outbursts, defined as in the case of observational data at a level 2 magnitudes

394 A. A. below maximum, and (2) the rates of decline following outburst maximum, determined at levels 1 and 2 magnitudes below maximum. The results are summarized in Table 1, where the widths W 2 are given in days, the rates of decline in days/mag, and the orbital periods in hours. We may recall that Models 11 18 correspond to hot = 0:1, while Models 21 28 to hot = 0:2. We should note at this point that those models were calculated for the simplest case of no enhancement in the mass transfer rate during outbursts. It is therefore important to recall that, as discussed in Paper II (Smak 1999), the rates of decline depend only very weakly on the amount of that enhancement, and the same is true for the widths of "narrow" outbursts (to be discussed in Section 5), corresponding to moderate enhancements. 4. The Problem of Disk Radii As discussed in Section 2, the existing correlations of the observed time scales with the orbital period simply reflect their strong dependence on the radius of the disk. It is therefore crucially important to make sure that the assumptions and procedures used in model calculations do indeed result in correct values of the disk radii. The radius of the disk, and its variations during the dwarf nova cycle, are obtained from models as a result of taking into account the following three effects: (a) the tidal removal of the angular momentum from the outer parts of the disk, (b) the deposition of the stream material (its specific angular momentum being lower than that of the disk outer parts), and (c) the outside transfer of the angular momentum within the disk. Concerning the tidal effects we should recall that Papaloizou and Pringle (1977) showed analytically that the effective tidal radius of the disk is: R tid 0:9R Roche (where R Roche is the mean radius of the Roche lobe), and that accordingly this became the standard assumption in model calculations (Smak 1984, 1998, Ichikawa and Osaki 1992, Hameury et al. 1998, 1999). In the case of dwarf novae the combined action of the three effects is to cause the disk expand during outburst and slowly contract during quiescence. Recent compilation of observational determinations and estimates of disk radii in eclipsing cataclysmic variables (Harrop-Allin and Warner 1996) shows, however, that generally they appear to be smaller than R tid 0:9R Roche. In particular, in the case of the best documented dwarf novae (U Gem and IP Peg) the radii of their disks vary during outburst cycle between R min 0:5? 0:7R Roche and R max 0:9R Roche (Smak 1996), to be compared with R min 0:8R Roche and R max 0:95R Roche, resulting from models listed in Table 1. These discrepancies are disturbing and require some explanation. An obvious possibility would be that our "prescription" for the tidal effects is incorrect or simply that R tid should be made smaller. Should this be the case, all model predicted time scales should also be revised! Fortunately, at least in the case of

Vol. 49 395 dwarf novae, we can perform an independent test on R tid, its results showing that there is no need for such a drastic revision. For a number of dwarf novae we have reliable determinations of their absolute visual magnitudes at the maximum of an outburst (Warner 1987) which obviously depend strongly on the radius of the disk. In what follows we shall concentrate on a sample of six dwarf novae with orbital periods in the range 3.8 4.6 h (see Table 1 in Warner 1987). The average value of their absolute magnitudes, corrected for inclination, is hm V (max)i = 4:47, with the three best systems (U Gem, CW Mon, and TW Vir) giving hm V (max)i = 4:40. We adopt this value for further discussion. For comparison we calculate M V (max) from two sets of models. Their basic parameters are: M 1 = 1:0M, M 2 = 0:4M, P orb = 4:2 h, and = 0:2 (values of M V (max) with = 0:1 are practically identical). Within a given set of models the free parameter is the mass transfer rate. The two sets of models are calculated with two different values of the tidal radius of the disk: the "standard" value of R tid = 0:9R Roche = 4:12 10 10 cm and a much lower value of R tid = 0:7R Roche = 3:20 10 10 cm. The results are shown in Fig. 1. Fig. 1. Absolute visual magnitudes at outburst maximum are plotted against the mass transfer rate. Upper set of points represents models with R tid = 0:9R Roche ; lower set with R tid = 0:7R Roche. Open squares correspond to Type B outbursts, filled squares to Type A outbursts. Vertical dotted line marks _M crit for R tid = 0:7R Roche. Arrow at right corresponds to the observed value of hm V (max)i.

396 A. A. The most important and rather obvious conclusion to be drawn from the comparison shown in Fig. 1 is that the absolute magnitudes predicted by models with R tid = 0:9R Roche agree with the observed hm V (max)i. This conclusion, however, must be supplemented with the following comments. (1) Fig. 1 shows that model values of M V (max) depend on the mass transfer rate. Fortunately, this dependence becomes significant only at the lowest values of M_ and, even there, the effect is not large enough to overshadow the much larger differences between R tid = 0:9R Roche and R tid = 0:7R Roche. In principle, this effect could be taken into account either by computing the effective mean absolute magnitude, or by adopting the mean mass transfer rate representative for the observed data sample. Neither can be done in a simple way. Qualitatively we can only conclude that with such a correction the resulting value of R tid would become slightly larger. (2) Models shown in Fig. 1 were calculated with no enhancement in the mass transfer rate during outbursts. In the case of moderate enhancements in _M tr we have M V (max) = 4:3? 4:4 (see Fig. 2 in Paper II), to be compared with M V (max) 4:4 (Fig. 1) for the case of no enhancement. Fortunately then the difference is only about 0.05 mag. (3) In the case of major enhancements in M _ tr which, as discussed in Paper II, are likely to be responsible for "wide" outbursts, maxima can become much brighter (see, again, Fig. 2 in Paper II). The observational data on the two types of outbursts (van Paradijs 1983) do indeed show such an effect with M V (max) = 0:2 mag. Since our models, intended to reproduce "narrow" outbursts, do not cover such situations, it is important to bear in mind that the observed values of M V (max), as listed for individual systems, are actually mean values, which may at least in some cases include "narrow" as well as "wide" outbursts. Consequently, our adopted observed value of hm V (max)i = 4:40 should be made slightly fainter, possibly by as much as 0.1 mag. Qualitatively we can conclude that taking into account effects (2) and (3) would make the resulting value of R tid slightly smaller. To summarize, it appears that in spite of these uncertainties we can conclude that R tid =R Roche 0:9, which is the standard value used in model calculations. Returning to the case of direct determinations of R d from eclipsing dwarf novae we should note that during outbursts, when the disk is large and therefore the tidal effects are very important, the model-predicted R max 0:95R Roche is roughly consistent with the observed R max 0:9R Roche. In fact, the agreement may actually be even better since no observational data are available close to the outburst maxima, when the eclipses of the spot are simply undetectable. It is only during quiescence that the model-predicted value of about R min 0:8R Roche appears to be significantly larger than R min 0:5? 0:7R Roche obtained from observations. There are two possible sources of this discrepancy, both of them unrelated to R tid. The first could be connected with the model description of the outside transfer of the angular momentum within the disk during quiescence. Should its efficiency be

Vol. 49 397 overestimated then the resulting radius of the disk would be larger. The second is connected with the very nature of the observational determinations of disk radii in dwarf novae, which are based on eclipses of the hot spot and an implicit assumption that the distance of the spot from the 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 due to that the structure of the hot spot may differ significantly from the simplified picture of a well defined "spot", located exactly at the outer edge of the disk. 5. The Width of Outburst vs. the Orbital Period Relation The observed widths (or durations) of outbursts are strongly correlated with the orbital period (cf. van Paradijs 1983, Szkody and Mattei 1984, Warner 1995). From a crude analysis of this correlation Gicger (1987) estimated 0:2. Our more detailed analysis below will confirm that value. Using our earlier conclusions concerning the nature of the "narrow" and "wide" outbursts (see Paper II), in what follows we shall discuss the width vs. the orbital period relation for "narrow" outbursts. Data are taken from van Paradijs (1983). It should be noted at this point that contrary to what is stated in the text the numbers listed in columns 10 and 11 of his Table 1 are not the reduced widths, corresponding to a level 2 mag below maximum (those are shown in his Fig. 2), but the original widths determined at V w (column 5). Consequently, using the procedure described by van Paradijs, we recalculate the reduced widths of "narrow" outbursts and plot them in our Fig. 2 (which is, obviously, identical with the upper part of Fig. 2 in van Paradijs paper). As can be seen the dependence on the orbital period is approximately linear: W = C W P orb ; (7) with formal fit to the data points, with P orb expressed in hours, giving C W = 1:39 0:06, and dispersion rms = 1:06. In a more general case, as suggested by analytical considerations (Eq. 6), the W? P orb relation could be written as W = C W 0 P orb : (8) Formal fit in this case gives: C W 0 = 2:01 0:29, = 0:78 0:11, with a dispersion rms = 0:96. Note that the value of compares reasonably well with that resulting from analytical considerations (Eq. 6). We should also note, however, that the rms dispersions in the linear and non-linear case are not significantly different. Therefore, in further discussion we shall simply use the linear approximation. Fitting model data from Table 1 with the linear relation (Eq. 7), we get C W (0:1)=2:620:10 for hot =0:1 and C W (0:2)=1:470:05 for hot =0:2. We may note that C W (0:1)=C W (0:2)=0:56 compares nicely with t(0:1)=t(0:2) = 0:62 from Eq. (6). Finally, by comparing the observed value of C W with the model

398 A. A. Fig. 2. The width of outburst vs. the orbital period relation for narrow outbursts. Crosses are observational data from van Paradijs (1983). Filled squares and triangles are model data with = 0:2 for Type A and Type B outbursts, respectively. values of C W (0:1) and C W (0:2), we get hot = 0:21 0:01. Accordingly, Fig. 2 shows model data with hot = 0:2. 6. The Rate of Decline vs. the Orbital Period Relation The observed rates of decline are strongly correlated with the orbital period (Bailey 1975, van Paradijs 1983, Szkody and Mattei 1984, Warner 1995). This relation was first discussed by Mantle and Bath (1983), who compared it with theoretical predictions based on a model, in which large enhancement in the mass transfer rate was assumed to be the only cause of dwarf nova outbursts. Their main result: 1 was certainly due to this assumption and other oversimplifications (note, in particular, that the shapes of their model light curves were clearly different from the observed ones). On the other hand, the first calculations based on the thermal instability model (Smak 1984) gave 0:2. Our present analysis, based on a larger data sample and a more representative set of models, will confirm that value. The observed rate of decline vs. orbital period relation is shown in Fig. 3,

Vol. 49 399 using data from Warner (1995, Tables 3.1 3.3). As in the case of outburst widths (Section 5), we begin by assuming a linear relation: = C P orb : (9) Formal fit, again with P orb expressed in hours, gives C = 0:38 0:02 with a dispersion rms = 0:54. This value of C is identical with the original value obtained by Bailey (1975) from a much smaller sample. Fig. 3. The rate of decline vs. the orbital period relation. Crosses are observational data from Warner (1995). Filled squares and triangles are model data with = 0:2 for Type A and Type B outbursts, respectively. In a more general case we can assume = C 0 P orb : (10) Formal fit in this case gives: C 0 = 0:61 0:07, = 0:71 0:09, with a dispersion rms = 0:48. These values are slightly different from those obtained by Warner (1995, Eq. 3.5), due mostly to the fact that in his fit he included objects with periods longer than 10 h. Again, as in Section 5, we note a good agreement with predicted by Eq. (6).

400 A. A. Turning to model data, and using the linear relation (Eq. 9), we get C (0:1) = 0:74 0:04 for hot = 0:1 and C (0:2) = 0:42 0:02 for hot = 0:2. Again we note that C (0:1)=C (0:2) = 0:57 compares nicely with t(0:1)=t(0:2) = 0:62 from Eq. (6). Finally, using these values and the observed value of C we get hot = 0:21 0:02. Accordingly, Fig. 3 shows model data with hot = 0:2. 7. Comparison between Outburst Widths and Rates of Decline We begin by recalling that the values of, obtained from the observational data for W and in the non-linear case (Eqs. 8 and 10), agree reasonably well with predicted by Eq. (6). The model predicted ratios C( = 0:1)=C( = 0:2) also agree in both cases with that predicted by Eq. (6). This implies that the concept of travel time, used in Section 2, can indeed be applied to the characteristic time scales observed during outbursts, and more importantly that this appears to be true regardless of the type of outburst (A or B). As could be expected from the existence of strong W? P orb and? P orb correlations, the observed outburst widths and rates of decline are correlated: = (0:292 0:013)W: (11) There is a scatter in all three correlations. After calculating deviations W and from the mean W? P orb and? P orb relations we find that those deviations are also correlated (with the correlation coefficient 0:7). This implies that the observed scatter is, at least in part, due to the intrinsic scatter in parameters other than P orb. One of them is the mass of the primary component. Using data available for M 1 we find, however, no correlation between W or and M 1 and this is consistent with the very weak dependence on M 1 predicted by Eq. (6). Another parameter, which could be identified as the possible source of the observed scatter, is M _. Unfortunately, as discussed in Section 2, its physical meaning is not quite obvious and this precludes any further discussion of this point. Finally, it is not possible that the scatter in the W? P orb and? P orb relations could also be due to. 8. The UV Delay As discussed in Paper I, the model-predicted UV delays agree with observations, for hot = 0:1? 0:2. Unfortunately, due to large observational scatter and a relatively small sample, a more precise determination of this parameter is impossible in this case.

Vol. 49 401 9. Discussion The most important and most obvious conclusion of the present paper is that the three characteristic time scales observed during dwarf nova outbursts give, consistently, hot 0:2. Another important conclusion, resulting from a comparison of the absolute visual magnitudes at maximum, and concerning the tidal radii of accretion disks in dwarf novae, is that R tid =R Roche 0:9. This proves that disk radii and, consequently, the characteristic time scales, resulting from model calculations using this "standard" value, are essentially correct. REFERENCES Bailey, J. 1975, J. Brit. astr. Ass., 86, 30. Gicger, A. 1987, Acta Astron., 37, 29. Hameury, J.-M., Menou, K., Dubus, G., and Lasota, J.-P. 1998, MNRAS, 298, 1048. Hameury, J.-M., Lasota, J.-P., and Dubus, G. 1999, MNRAS, 303, 39. Harrop-Allin, M.K., and Warner, B. 1996, MNRAS, 279, 219. Ichikawa, S., and Osaki, Y. 1992, Publ. Astr. Soc. Japan, 44, 15. Kurucz, R.L. 1993, CD-ROM No. 13. Mantle, V.J., and Bath, G.T. 1983, MNRAS, 202, 151. Papaloizou, J., and Pringle, J.E. 1977, MNRAS, 181, 441. Ritter, H., and Kolb, U. 1998, Astron. Astrophys. Suppl. Ser., 129, 83. Smak, J. 1984, Acta Astron., 34, 161. Smak, J. 1992, Acta Astron., 42, 323. Smak, J. 1996, Acta Astron., 46, 377. Smak, J. 1998, Acta Astron., 48, 677 (Paper I). Smak, J. 1999, Acta Astron., 49, 383 (Paper II). Szkody, P., and Mattei, J.A. 1984, P.A.S.P., 96, 988. van Paradijs, J. 1983, Astron. Astrophys., 125, L16. Warner, B. 1987, MNRAS, 227, 23. Warner, B. 1995, "Cataclysmic Variable Stars" (Cambridge Univ. Press). Webbink, R.F. 1990, "Accretion-Powered Compact Binaries", Ed. C.W. Mauche (Cambridge Univ. Press).