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

2. The ²⁶Al abundance in the ejecta of O-Ne-Mg novae

In order to calculate the amount of ²⁶ Al produced by ONeMg novae in the Galaxy, we need to know how much ²⁶ Al is produced by an ONeMg nova of a given set of parameters and then how such novae are distributed over those parameters. We discuss the latter point in Sect. 3.

2.1. Computation of a production function

To determine the mass fraction of ²⁶ Al in the ejecta of ONeMg novae, we use the results of recent hydrodynamic studies of accretion onto massive (ONeMg) WDs by Politano et al. (1995, 1996). These computations use an implicit, 1-D hydrodynamic stellar evolution code to follow the TNR. The code is coupled to an extended nuclear reaction network, including 78 nuclei ranging from H to ⁴⁰ Ca. The nuclear network is described in Politano et al. (1995) and Weiss & Truran (1990). The hydrodynamic code to which the network is coupled has been described extensively in the literature (e.g., Kutter & Sparks 1972; Starrfield & Sparks 1987). We refer the reader to these papers for further details.

Three sets of model sequences were calculated by Politano et al. (1995, 1996). In the first set of sequences, the dependence of the outburst on the mass of the WD was investigated. In the second set of sequences, the dependence of the outburst on the amount of enrichment of the accreted material in O, Ne, and Mg was investigated. Finally, in the third set of sequences, the dependence of the outburst on the accretion rate was investigated. For a given set of sequences, only one parameter was varied, the other two remaining fixed. In all three sets, an initial WD luminosity of was used. Initial model parameters and ²⁶ Al mass fractions in the ejecta for all three sets of sequences are shown in Table 1, together with an estimate for the total mass of ²⁶ Al ejected per nova event by the corresponding model.

Table 1. ²⁶ Al and ²² Na abundance, and , in the ejecta of ONeMg nova models for different white dwarf masses , enrichments d of the accreted material in O, Ne and Mg, and mass accretion rates . The sixth column provides an estimate for the total mass of ²⁶ Al ejected into the interstellar medium per TNR for the corresponding nova model. Eq. (12) was used to estimate the total ejected mass.

Here, we model the mass fraction of ²⁶ Al in the ejecta of ONeMg novae by assuming it can be approximated by a product function of the three parameters investigated by Politano et al. (1995, 1996). Accordingly, we obtain as a production function

where is the mass of the WD, is the accretion rate, and is the ratio of the mass ejected in a given nova outburst to the mass accreted by the WD between outbursts. We have normalized the factors , and to unity for the nova model with ,  gs^-1, and (which we will hereafter refer to as our "standard" nova model). The normalization factor, , is then simply the ²⁶ Al abundance in the ejecta of this standard model (see below). Before discussing the individual functions in (1), we first discuss the parameter , and its relationship to the assumed amount of enrichment of the accreted material in O, Ne, and Mg.

The presence of significant enrichments of CNO or intermediate-mass elements in the ejecta of most classical novae suggests that the WD mass is being eroded. Such significant amounts of enrichment are very difficult to understand unless material from the underlying WD is being mixed into the accreted envelope material (e.g., Truran 1990). Unless essentially all of the enriched envelope (i.e., accreted hydrogen-rich companion material and dredged-up WD material) is ejected during the explosion, there would remain on the WD some hydrogen-rich material. During the constant bolometric phase, this hydrogen would be burned to helium. If this process continued, then, over the course of several outbursts, eventually the composition of the underlying WD would be masked by the build-up of remnant helium in the envelope. Since we do observe substantial enrichments of CNO or intermediate-mass elements in the ejecta of novae, we conclude that not only must an amount of mass equal to the accreted companion mass be ejected, but also an amount of mass equal to the amount of WD material that was mixed into the envelope. This gives us a relationship between , the ratio of the ejected mass, , to the accreted mass, , and d, the amount of enrichment (by mass) of the accreted material in O, Ne, and Mg:

In order to construct the functions, , , and , we have made analytic fits to the data in Table 1 for the separate dependencies on , , and . These analytic fits are given below and the corresponding functions, along with Politano et al.'s data, are shown in Fig. 1.

where , and

Fig. 1. a (upper panel): 26 Al abundance in nova ejecta as a function of WD mass, , computed according to (1) with . The curves correspond to , -8.8 and -7.8 (from bottom to top); b (lower panel, left): from Eq. (4); c (lower panel, right): from Eq. (5). Crosses mark the results from actual model calculations by Politano et al. (1995, 1996), see Table 1.

In the enrichment and accretion rate sequences the pre-runaway abundances were computed differently than in the WD mass sequences. As a result of this, the ²⁶ Al mass fraction is 40% higher in the 50% enrichment sequence than in the 1.35 WD mass sequence, even though these two sequences were computed for identical sets of parameters (1.35 , 50% enrichment, 10¹⁷ g s^-1). To compensate for this inconsistency between the WD mass sequences and the rest of the sequences, we have increased all of the ²⁶ Al mass fractions in the WD mass sequences listed in Table 1 by 40%. In particular, we adopt for the normalization in (1). We note that Hernanz et al. (1996) investigated the TNR on a ONeMg WD using similar techniques and input physics as Politano et al. (1995, 1996) and find an ²⁶ Al output consistent with (1).

One of the central factors which determines the behavior of , , and is the peak burning temperature achieved during the runaway. High peak temperatures (T 100 10⁶ K) are needed to produce the ²⁶ Al via the reaction sequence: ²⁴ Mg(p, )²⁵ Al()²⁵ Mg(p, )²⁶ Al. However, once the temperature exceeds 200-250 10⁶ K, destruction of ²⁶ Al via the reaction, ²⁶ Al(p, )²⁷ Si, competes favorably with production mechanisms (e.g., Nofar et al. 1991; Politano et al. 1995). In the case of the WD mass sequences, as the WD mass increases, the outburst becomes more violent. The peak temperature increases from 224 K in the 1.00 sequence to 356 K in the 1.35 sequence. Correspondingly, the mass fraction of ²⁶ Al decreases monotonically by a factor of . In the enrichment sequences, it may first appear unusual that the mass fraction of ²⁶ Al doesn't increase monotonically as the pre-runaway envelope becomes more enriched in O, Ne and Mg. After all, it is proton captures on ²⁴ Mg which are ultimately responsible for the production of ²⁶ Al during the runaway. However, as the enrichment in O, Ne, and Mg increases, the mass fraction of carbon decreases. Proton captures on ¹² C are responsible for initiating the runaway. Therefore, a reduction in the ¹² C abundance in the envelope delays the runaway, and this allows the envelope material to become more degenerate. Once the runaway occurs, the higher degeneracy causes the outburst to be more violent and higher peak temperatures are reached. Peak temperatures increase from 257 10⁶ K in the 25% enrichment sequence to 390 10⁶ K in the 75% enrichment sequence. Thus, there are two competing effects as the enrichment is increased: more ²⁴ Mg is available, which means that more ²⁶ Al can be produced, but higher peak temperatures are achieved, which means that more ²⁶ Al can be destroyed. These competing effects result in a maximum in f() near (50% enrichment). Finally, in the accretion rate sequences, as the accretion rate decreases, longer and longer periods of time are required in order to achieve a runaway. This allows the accreted material to become more and more degenerate, and the ensuing runaway more and more violent. This effect is very prominent; peak temperatures increase to 700 10⁶ K in the 10¹⁶ g/s sequence. The mass fraction of ²⁶ Al in the ejecta shows a correspondingly sizable decrease.

Several factors in the model calculations can cause uncertainties in the corresponding mass fraction of ²⁶ Al. First, the initial WD temperature in all of the models is probably too high. Studies of WD temperatures in dwarf novae soon after outburst suggest that the WD is cooler than expected (Long et al. 1994). Second, the opacities are still uncertain. Even with the newer (OPAL) opacities, there are still some problems. In particular, because the material is partially degenerate, at certain points in the evolution extrapolations must be made to the opacities where the temperature and density values are outside of the range of the tables. Third, modeling convection is problematic since the nuclear burning time scale can be comparable to or shorter than the convective turn-over time. As with all other 1-dim. nova simulations that we are aware of, we use a mixing length description for convection, adapted to the physical situation under consideration as described in Starrfield et al. (1978). Convective mixing is handled by solving the diffusion equation at the end of each time step for each isotope over the convective region. A closer inspection of the interplay between convective turbulent motion and nuclear burning similar to the case of Type Ia supernovae (e.g. Niemeyer & Hillebrandt 1995) would require a 2-D or 3-D hydrodynamic model which is beyond the scope of this study. Nevertheless, we believe that the implemented procedure represents a satisfactory description of convection and convective mixing in the context of a TNR in a classical nova. Fourth, the luminosity due to accretion was not included in the models. Fifth, and probably most importantly, while the most up-to-date reaction rates were used at the time of the calculations, some of the rates particularly relevant to ²⁶ Al production were uncertain. More accurate reaction rates have been published recently by Herndl et al. (1995). Very recent calculations similar to Politano et al. (1995) have been performed by Starrfield et al. (1996) using the newer rates. A comparison for a 1.25 WD sequence suggests that the ²⁶ Al mass fraction may be reduced by as much as a factor of , although one must be careful in making a direct comparison since other factors (e.g., opacities, WD temperature) were also varied.

2.2. A parameterized production function

In view of these uncertainties, we wish to investigate the importance of each of the factors , , and individually for the predicted overall Galactic ²⁶ Al production. To do this, we consider an artificial production function similar to (1), but containing adequately chosen free parameters. The first free parameter is the absolute calibration of the standard model. Since this represents only an overall scaling factor, its actual value has no influence on our differential study, and for consistency we simply use the same value as in (1), i.e., . From the considerations in the previous section, we expect that any production function obeys similar differential trends as expressed in Eqs. (3)-(5), i.e. that decreases with increasing WD mass, increases with increasing accretion rate, and that reaches a maximum value for an intermediate degree of envelope/core mixing values ( in the vicinity of ). Hence, we adopt

as a generalized form for , with as a free parameter. The functional dependence expressed in (3) corresponds roughly to ; in Sects. 4 and 5 we vary , the steepness of the decrease in with increasing WD mass, from 0 to 8.

To find a generalized , we first note that within standard models for CV evolution the mean mass transfer rate in CVs is in the range for long-period systems above the so-called CV period gap (  h), and for short-period CVs below the gap (  h; see e.g., Kolb 1993a). If the accretion rate is on the average the same as the transfer rate, then (4) suggests that the contribution of systems below the gap is negligible. Detailed population models (see the discussion in Sect. 5.1) with the production function derived in Sect. 2.1 confirm this expectation. Assuming the same property for the generalized , we use the expression

with as a free parameter. Eq. (4) is equivalent to . In Sects. 4 and 5, we explore the influence of , which measures how steeply increases with increasing accretion rate, on our results as is varied between 0 and 8.

Finally, we note that plays no role in the differential comparison presented below. We assume that has a fixed value, , for a given nova population model (i), and the amount of ²⁶ Al produced scales linearly with .

Results of model computations using this generalized production function are shown in Sect. 4 and discussed in Sect. 5.2.4.

© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998
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