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Astron. Astrophys. 319, 909-922 (1997)

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5. Discussion

A main goal of the present study is to derive limits on the 26 Al production by ONeMg novae from standard models for the formation and evolution of CVs, and explore to what extent the observed upper limit of [FORMULA] of 26 Al in a homogeneous background puts constraints on free parameters in these standard models.

From our main results, which are depicted in Figs. 2 and 3, we first note two obvious trends:

1) Within a given set of models the amount of 26 Al produced increases with increasing [FORMULA] as long as [FORMULA], where [FORMULA] reaches a maximum (Fig. 1c). Hence we expect that the curves in Figs. 2 and 3 for [FORMULA] are close to an upper limit for the 26 Al production in the corresponding population. This is true even if [FORMULA] is not a constant but depends on system parameters. 2) A comparison of the models in sets A and B which belong to the same value of [FORMULA] shows that models from set B produce typically 5-10 times more 26 Al than models from set A, consistent with the larger number of systems in set B.

Moreover, from Fig. 1b it can be seen that CVs with high mass transfer rates dominate the 26 Al production not only via the factor [FORMULA] in Eq. (8), but also due to the relative preference of these systems by [FORMULA]. Therefore, the 26 Al is almost exclusively produced by systems above the period gap. Note that this is a non-trivial result: although most classical novae are indeed found to have periods above the period gap, these systems comprise only [FORMULA] of the total intrinsic population (Kolb 1993a). Long-period nova systems have a greater detectability since the mass transfer rate, and consequently the outburst frequency, is a factor 10 to 100 higher than for short-period ones. The mass transfer rate above the gap, on the other hand, can't be very different from the values obtained from the specific formulation of magnetic stellar wind braking used here, since the width of the period gap alone requires that [FORMULA] at the upper edge of the gap (Stehle, Ritter & Kolb 1996, see also Kolb 1995b, 1996; this is one of the most robust predictions of the standard model of CV evolution), suggesting that the precise form of the magnetic braking is to zeroth order not important. We also note that the 26 Al production curve obtained from a test calculation with a smaller common envelope efficiency ([FORMULA]) was very close to the curve for [FORMULA], confirming again the finding by Kolb (1993a) that [FORMULA] plays only a minor role in determining the present CV population.

5.1. Population models

Before proceeding further, it is desirable to test predictions from the CV population models we are using against observations not related to 26 Al. These independent tests will allow us to better constrain some of the uncertain parameters, particularly [FORMULA] and [FORMULA], which enter into our investigation of 26 Al production from ONeMg novae.

One such test is to compare a visual magnitude-limited sample drawn from the computed population with the observed orbital period distribution of (non-magnetic) CVs, thereby neglecting further selection effects (a more detailed discussion is given in Kolb 1996). Population models with small or no correlation between the component masses in main-sequence binaries, similar to models of set B, tend to agree better with the observed fraction of CVs found below the period gap. Such models are also favored by a comparison of observed post-CE binaries (detached WD/main sequence stars, binary central stars of planetary nebulae, binary subdwarf O and B stars, barium stars) with calculated distributions obtained from the same initial set of main-sequence binaries (e.g. deKool & Ritter 1993).

A second test is a comparison of the observed mean WD mass in CVs ([FORMULA], e.g. Ritter & Kolb 1995) with the value calculated from the above-mentioned visual magnitude-limited samples. As was pointed out by Ritter & Burkert (1986), and confirmed by Dünhuber 1993 (see also Dünhuber & Ritter 1993), selection effects in favor of the high-mass WDs are responsible for the fact that the observed mean value is significantly above the mean WD mass observed in isolated WDs ([FORMULA], Schmidt & Smith 1995). Dünhuber finds no clear preference for a particular population model. However, his investigations were based on models where the WD mass was held constant during the secular evolution of CVs, i.e [FORMULA]. If [FORMULA] is different from 1, as is the case in our models, the mean WD mass becomes a function of [FORMULA]. Kolb (1993b) derives [FORMULA]. In this case, the computed mean WD mass is sufficiently close to the observed value only for [FORMULA]. We note that this limit applies only for a global value of [FORMULA], that is, only under the assumption that [FORMULA] is the same for the entire population. However, [FORMULA] may depend on the mass of the WD and therefore the mean [FORMULA] may be different for different subsets of the population (e.g., for CVs with CO WDs as opposed to ONeMg WDs; see the discussion in Sect. 5.2.3. below).

Lastly, we test our population models against observations of novae. As can be seen from Table 2, models of set B predict a nova rate that is roughly 10 times higher than models of set A, reflecting the different number of CVs in the population. The nova rate is only a weak function of [FORMULA], despite the apparent proportionality to [FORMULA] in Eq. (10). This can be understood from Eq. (11) which shows that high-mass WDs contribute quite significantly to the nova rate since [FORMULA] formally approaches 0 when the WD mass gets close to the Chandrasekhar mass. In populations with [FORMULA], the WDs shrink in mass throughout their evolution, reducing the high-mass WD contribution to [FORMULA]. Using (12) instead of (11) results in essentially the same Galactic nova rate, confirming that the prescription given in (11) reproduces the ignition mass fairly well (except for WDs very close to the Chandrasekhar mass). The extrapolation of (12) to WD masses much smaller than [FORMULA] is of course without physical significance and yields ignition masses a factor [FORMULA] higher than (11) for [FORMULA]. This failure for low-mass WD systems does not propagate into the predicted nova rate since that is completely dominated by higher-mass WD CVs. The nova rates computed according to Prialnik & Kovetz (1995) lie generally slightly above the rates computed with the canonical ignition mass and this difference increases with increasing [FORMULA]. However, we note that deducing the nova rate from our population models using (13) as the ignition criterion introduces an inconsistency since the underlying nova models for this criterion do not belong to a constant value of [FORMULA], rather [FORMULA] depends on the corresponding WD mass, mass accretion rate and WD temperature (typically [FORMULA], except for very high accretion rates [FORMULA] where [FORMULA]). Indeed, the motivation to use an ignition condition based on these latter models at all is to demonstrate that the predicted Galactic nova rate is not very sensitive to the precise ignition condition, so that, in this context, (11) is a satisfactory approximation for it.

Concerning this last test, we note that a direct comparison with the value deduced from observations of 20 novae per year in the Galaxy must be taken with care because of the considerable uncertainties in both the computed and observed values. Nevertheless, models with uncorrelated main-sequence masses and small values for [FORMULA] seem to match the observations best. Given the fact that the computed nova rate is not very sensitive to the ignition criterion, the predicted total nova rate may well prove to be a major constraint on the models. We, therefore, emphasize the need for more reliable observational determinations of the nova rate in our Galaxy.

5.2. Predicted 26 Al production

In the light of the above considerations, we proceed to discuss limits and constraints that can be placed on the 26 Al production in ONeMg novae.

We begin by first discussing estimates for global upper and lower limits to 26 Al production in ONeMg novae. We then discuss likely regimes for 26 Al production. Finally, we close this section by discussing the effects of using the parameterized production function (see Sect. 2.2) on our results.

5.2.1. Global upper limit

Since models of set B produce more 26 Al than models of set A, we use set B to establish our upper limit. Further, since [FORMULA] reaches a maximum near [FORMULA], we can use the [FORMULA] curve in Fig. 3 to estimate an upper limit to 26 Al production as a function of [FORMULA] for this population. Inspection of Fig. 3 shows that if [FORMULA], the resulting upper limit surpasses the COMPTEL upper limit of [FORMULA] 26 Al in the homogeneous background. To establish a global upper limit for 26 Al from ONeMg novae, we must eliminate the dependence on [FORMULA] by finding a reasonable value for [FORMULA]. As we have said, stellar and binary evolution calculations are of little help in this regard. Instead, we will use a comparison of the observed fraction of ONeMg novae with our predicted fraction, [FORMULA], for this purpose. While we recognize the uncertainties in both the observed and predicted values of [FORMULA] (see Sect. 3.2), we feel that this will give at least a reasonable estimate for [FORMULA]. We also note that advantages of this approach are that our predicted [FORMULA] is insensitive to [FORMULA] and, since it is a relative quantity, we avoid any uncertainty related to the normalization of a population model. Choosing [FORMULA] = 0.3 to agree with observations (see Sect. 3.2), we find [FORMULA]. From Fig. 3, this translates into an estimate of [FORMULA] ([FORMULA]) as a global upper limit for 26 Al production from ONeMg novae.

5.2.2. Global lower limit

While the property of [FORMULA] that it reaches a global maximum near [FORMULA] allowed us to establish reasonably straightforwardly an estimate for a global upper limit, the situation is more complex for a global lower limit. As we discussed in Sect. 2.1, observed heavy element enrichments in nova ejecta almost inescapably suggest that [FORMULA] for most classical novae: the question is how much greater? Population models which take into account a secular decrease of the WD are only able to constrain a mean or global value of [FORMULA] for the population to the range [FORMULA]. Ejecta abundances determined from observations, even in well-studied systems, are fraught with uncertainties and show a large scatter in Z across the board (i.e, for both CO and ONeMg systems; see Starrfield et al. 1996). While these abundance determinations may be used with some confidence to distinguish CO novae from ONeMg novae, they do not allow a firm determination of [FORMULA] for each system. Finally, the gist of the problem with quoting a lower limit really lies with our uncertain knowledge of mixing as we discussed in Sect. 3.3. Until our understanding of the physical mechanism responsible for mixing in classical novae improves, it will be very difficult to quote a global lower limit to 26 Al production in ONeMg novae (other than zero) with any confidence.

5.2.3. Constraints on 26 Al production

In this section we will use observational and theoretical constraints to attempt to establish likely values for 26 Al production in ONeMg novae. The observational constraints we use are: (1) the amount of 26 Al from diffuse sources based on the recent COMPTEL observations; (2) the fraction of ONeMg novae among novae that have been observed in outburst; and (3) ejecta abundances determined from well-studied novae. Before proceeding further, a few words about the uncertainties in these quantities is in order. The value of [FORMULA] for 26 Al from diffuse sources is probably the most secure of the three. While a somewhat higher value cannot be excluded and while some ONeMg novae may possibly be associated with regions of isolated emission, using [FORMULA] to place an upper limit on the amount of 26 Al from ONeMg novae is probably correct to within less than a factor of 2. While the current value of 30% for the observed fraction of ONeMg novae is a bit more uncertain, it has only fluctuated between [FORMULA] 25% and 50% (the latter including systems with marginal neon abundances) as more and more ONeMg novae have been observed over the past ten years. We therefore would estimate that the observed value for [FORMULA] is good to within a factor of 2. However, there is a further uncertainty: how to translate the observed value of [FORMULA] into the quantity [FORMULA] we predict theoretically. Our models properly take into account the main reason for the prevalence of ONeMg novae - the higher outburst frequency on high-mass WDs (e.g., Ritter et al. 1991, Kolb 1995a). But there may be additional selection effects favoring the observation of ONeMg novae: nova outbursts on high-mass WDs are likely to be brighter than outbursts on lower-mass WDs (e.g. Livio 1994). This would imply that our predicted [FORMULA] corresponds to somewhat lower values of the observed [FORMULA]. Finally, the most uncertain of the three constraints is the abundances of the ejected material in novae. While independent determinations of the ejecta abundances for the same system agree quite well in some cases (e.g., V1668 Cyg, V1370 Aql), there is disagreement in other cases, for the same system, by factors of 4 to 5 (e.g., QU Vul, PW Vul), making precise constraints from a single system unwise (see Starrfield et al. 1996). Instead, in the following we will use trends common to several systems or trends which require less precision (e.g., such as distinguishing ONeMg novae from CO novae) to place constraints.

The greater agreement between observations and predictions of post-CE and CV population models with weakly-correlated or uncorrelated progenitor primary and secondary masses, as discussed in Sect. 5.1, leads us to favor using models of set B to estimate likely values of 26 Al production from ONeMg novae. Fig. 3 shows that a value of [FORMULA] corresponds to [FORMULA]. Assuming a constant or mean value of [FORMULA] for the entire population, as we have done in this study, we may use the constraint [FORMULA], in order to remain consistent with the mean WD mass in CVs as derived from observations (see Sect. 5.1). Looking at the [FORMULA] curve in Fig. 3, for [FORMULA], we have a corresponding 26 Al production of [FORMULA].

We emphasize that this value depends greatly on the assumption that [FORMULA] is the same for the entire population. If, for example, the mean [FORMULA] in ONeMg novae is higher than the mean [FORMULA] for all novae, then a larger 26 Al production is allowed. In order for ONeMg novae to account for the entire 1 [FORMULA] of 26 Al in a diffuse background implied by the COMPTEL observations, models of set B require that the mean [FORMULA] in ONeMg systems be [FORMULA] (for [FORMULA]). This implies a mean mixing of [FORMULA] 40% in ONeMg systems, as compared with a maximum mean mixing of [FORMULA] 20% for the entire population implied by the constraint of [FORMULA] discussed above. We explore below to what extent a higher degree of mixing in ONeMg novae is supported by ejecta abundances in novae.

Inspection of ejecta abundances in well-studied novae (see Table 1 in Starrfield et al. 1996) reveals that the mean Z in systems which are unambiguously ONeMg novae ([FORMULA]) is 0.52 compared with a mean Z for all well-studied novae of 0.34. Care must be taken in interpreting this point, however, since in order for a system to be unambiguously considered as a ONeMg nova, the enrichment in neon over solar must be well in excess of 10 [FORMULA] solar (see Sect. 3.2), biasing the mean Z in ONeMg nova systems to be high. If one includes novae with [FORMULA] (V977 Sco, V2214 Oph and V1500 Cyg), then the mean Z is reduced to 0.42. Nevertheless, it does appear that the neon abundances in two ONeMg novae are quite high (V693 CrA, [FORMULA] and V1370 Aql, [FORMULA] ; we note that two independent abundance determinations were made for each system and the values for [FORMULA] agree to within 40% for V693 CrA and to within 10% for V1370 Aql.) Comparison with the nova models of Politano et al. (1995, 1996) suggests that the level of mixing is of order [FORMULA] or greater for these systems.

Finally, we close by noting that the null detection of 22 Na [FORMULA] -ray emission from nearby ONeMg novae (Iyudin et al. 1995) imposes constraints on the amount of 22 Na produced in ONeMg novae. The authors estimate an upper limit for the ejected 22 Na mass for such novae within the Galactic disk of 3.7 [FORMULA] 10 [FORMULA]. However, this constraint does not necessarily restrict [FORMULA] to small values. Politano et al. (1996) produce models with very high mass WDs (1.35 [FORMULA]) where the 22 Na mass fraction is low even for moderate values of [FORMULA] (of order 50%), cf. last column of Table 1. Nova outbursts on lower mass ONeMg WDs [FORMULA] are not expected to produce significant amounts of 22 Na because the peak temperature is too low (e.g., Politano et al. 1995). In addition, the unusually high ejecta masses in a number of ONeMg novae (e.g., QU Vul [Taylor et al. 1988], V838 Her [Woodward et al. 1992]), V1974 Cyg [Pavelin et al. 1993]) pose difficulties for standard models of nova outbursts on high-mass WDs, which predict much lower ejecta masses (e.g., Starrfield 1989; Politano et al. 1995). These latter two points make it imperative for observers to obtain reliable WD masses for ONeMg nova systems, particularly those with high ejected masses (see the discussion in Misselt et al. [1995]). If such determinations reveal WD masses of [FORMULA] in these systems, then either alternative mechanisms for producing ONeMg-rich WDs in binary systems with lower-mass WDs ([FORMULA] at birth) must exist (e.g., Shara & Prialnik 1994; Shara 1994), or the mean amount of mixing in ONeMg novae is significantly higher than for novae in general, leading to substantial erosion of the WD during the system's lifetime. Either case would have important implications for 26 Al production in novae.

5.2.4. Effects of the parameterized production function on 26 Al production

Our motivation for using a parameterized production function is to attempt to estimate the effects of the uncertainties in the nova models on 26 Al production (see Sects. 2.1 and 2.2). Recall from Sect. 2.2 that we generalized the production function in Sect. 2.1 (which we will hereafter refer to as our "standard" production function), while keeping the same differential trends, using simple parameters, [FORMULA] and [FORMULA], to explore a greater range of dependencies on [FORMULA] and [FORMULA]. In particular, [FORMULA] and [FORMULA] were varied from 0 to 8, probing stronger and stronger dependencies on WD mass and accretion rate, respectively. Our standard production function corresponds approximately to [FORMULA] and [FORMULA]. Thus, for [FORMULA], we have a weaker (flatter) dependence on WD mass than in our standard production function for [FORMULA] (Eq. 3), whereas [FORMULA] corresponds to a stronger (steeper) dependence on WD mass. Similarly, for [FORMULA], we have a weaker (flatter) dependence on accretion rate than in our standard production function for [FORMULA] (Eq. 4), whereas [FORMULA] corresponds to a stronger (steeper) dependence on accretion rate.

The results of using the parameterized production function are shown in Fig. 4. As noted previously, the 26 Al production in this figure is normalized to [FORMULA], the 26 Al production in our "standard" model. The qualitative dependencies on [FORMULA] and [FORMULA] expressed by these curves are as expected from the functional form of the parameterization: namely 26 Al production increases with both increasing [FORMULA] and [FORMULA]. It is interesting, however, to note that the influence of [FORMULA] is rather modest (a factor of 2 change at most), whereas the influence of [FORMULA] can be substantial (factors of 2-30) especially for larger values of [FORMULA]. This shows that the influence of the accretion rate dependence is weaker than the influence of the WD mass dependence on 26 Al production for a given population.

Of the uncertainties in the nova models discussed in Sect. 2.1, the greatest concern is the potential influence of the new reaction rates described in Herndl et al. (1995) on 26 Al production in novae, and we would like to discuss this briefly here. Since the dependence on [FORMULA] is small (at least of the same order as the uncertainties in the observational quantities we used to place constraints on 26 Al production in the first place), we focus on how the new rates may affect the dependence on [FORMULA]. Network calculations by Herndl et al. (1995) show that the new reaction rates result in a decreased 26 Al production compared with the rates used by Politano et al. (1995, 1996). However, Herndl et al. point out that the new rates will only affect the production of 26 Al for temperatures greater than [FORMULA] 106  K. From the WD mass sequence (Politano et al. 1995), this peak temperature corresponds to a WD mass between 1.00 [FORMULA] ([FORMULA]  K) and 1.25 [FORMULA] ([FORMULA]  K). This will tend to steepen the dependence on WD mass compared with the standard production function based on Politano et al.'s data, so that [FORMULA] should be somewhat larger than 3. However, the new rates will not only affect the differential trends with WD mass, accretion rate, etc., but also the absolute amount produced by a given model. In particular, therefore, the value of [FORMULA] (the amount of 26 Al produced from our standard model) will presumably be smaller and hence the normalization used in Fig. 4 be different. While preliminary calculations using the new rates performed by Starrfield et al. (1996) on a 1.25 [FORMULA] WD indicate that the 26 Al production is reduced by a factor of [FORMULA], we emphasize that several other factors (such as the initial WD luminosity, opacities, etc.) were also varied compared with Politano et al.'s models, so that one cannot directly infer that the effect of the new rates was of the same amount. Even assuming it were, the drop by a factor of [FORMULA] would be weakened by the effect of the simultaneous increase of [FORMULA]. Hence this seems to suggest an overall modest reduction of the 26 Al output in models with the new rates. However, we cannot make a definitive statement until detailed comparison models which use the new rates are calculated for several WD masses.

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Online publication: July 3, 1998