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

3. Nova population models

In addition to knowledge of how much ²⁶ Al is produced by an ONeMg nova of a given set of parameters, an estimate of the total amount of ²⁶ Al produced by ONeMg novae in the Galaxy requires knowledge about how novae are distributed in the Galaxy. Previous work (Weiss & Truran 1990) relied on observational estimates of the total Galactic nova rate and could not account for the dependence of on nova parameters as expressed in Eq. (1). A first step towards a more consistent estimate for the overall ²⁶ Al production in novae was attempted by Politano et al. (1995) who combined their findings for the mass dependence of with a theoretically computed WD mass distribution in novae (Ritter et al. 1991; Politano 1996a).

In our study, we extend this procedure by using a much more detailed, theoretically-predicted Galactic population of classical novae which has been derived from population models for CVs. Such population models (Kolb 1993a) essentially describe the distribution of the present-day Galactic population of CVs in a 4-dimensional configuration space defined by WD mass , secondary mass , mass transfer rate and orbital period P and, therefore, provide self-consistently the framework necessary to take advantage of the full, explicit dependencies of on and expressed in Eq. (1).

3.1. Model assumptions for CV populations

We emphasize at this point that our calculations are based on the widely-accepted standard models for formation and evolution of CVs (e.g. King 1988, Politano 1996a, Kolb 1995b, 1996). Although we test the influence of certain parameters on our results within this standard model, no attempt is made to introduce non-standard effects during or immediately after the common envelope phase (e.g. Terman & Taam 1996), or frictional angular momentum loss during the CV evolution (Schenker, Kolb & Ritter 1996; Kolb et al. 1996). We briefly summarize the physical assumptions and techniques used to calculate the CV model population below.

Main-sequence binaries are assumed to form continuously at a constant rate according to certain given distributions of the primary mass , the mass ratio ( being the secondary mass, ) and the orbital separation. Applying simple analytical fits of single star evolutionary calculations, their evolution is followed up to the point when the primary fills its Roche lobe for the first time. In CV progenitors at this stage the primary is a giant, while the secondary is still an unevolved, low-mass main-sequence star. The ensuing mass transfer is dynamically unstable, leading to a common envelope (CE) phase during which the orbital separation is reduced considerably. Released orbital energy is consumed to eject the primary envelope, leaving behind a WD (the exposed core of the giant), and the secondary, which is basically unaffected by the CE evolution. The post-CE orbital separation is computed in the usual way with the common envelope efficiency, , set to 1 (we use the definition of given in deKool 1992). Gravitational radiation and magnetic stellar wind braking shrink the orbit of the post-CE binaries further, bringing the secondary into contact with its Roche lobe, and drive the subsequent semi-detached CV evolution. Magnetic braking is computed according to Verbunt & Zwaan (1981). The braking is calibrated so as to reproduce the width of the CV period gap, and is assumed to be effective only as long as the secondary retains a radiative core. An isotropic stellar wind from the WD is used to approximate the effect of nova outbursts on the long-term evolution of CVs. This wind carries the WD's specific angular momentum and a mass per unit time, , so that , where is the parameter introduced in Eq. (2) and is assumed to be constant.

To create the population models, we started from calculations for the formation rate of newborn CVs by deKool (1992) and Politano (1996a) and computed the present, evolved state of the CV population with the CV population synthesis technique described by Kolb (1993a). In this procedure it is assumed that the CV formation rate has remained constant during the past yr, the assumed age of the Galaxy. Test calculations in which the explicit time dependence of the CV formation rate was taken into account show that the error introduced by this assumption is small, and altogether negligible for a calculation of ²⁶ Al production since the relevant high-mass WD CVs are the first in the CV population to reach a stationary state (e.g. Politano 1996a). The evolved population is obtained by combining a large number of CV evolutionary sequences computed with the generalized and calibrated bipolytrope code (Kolb & Ritter 1992) which cover the initial configuration space on a sufficiently dense grid. The sum over all initial configurations is then weighted according to the formation rate and integrated over the star formation history in the Galaxy (e.g. Kolb 1993a).

Of the parameters which enter into a calculation of the formation rate of CVs, the initial distribution of the mass ratio, q, in main-sequence binaries is the most controversial, yet is the one with the most significant influence on the final CV population (e.g., deKool 1992; Kolb 1993a; Politano 1994). Thus, we test two somewhat extreme cases, one with a very strong correlation between primary and secondary mass, and one without any correlation at all (see Sect. 4).

The ratio of the ejected mass to the accreted mass, specified by the global parameter in Eq. (2), is the key parameter needed to compute the amount of ²⁶ Al produced by the corresponding CV population. The mass transfer rate and the ²⁶ Al mass fraction from Eq. (1) then determine the mass of ²⁶ Al a given system ejects per unit time:

Assuming that ²⁶ Al has an equilibrium abundance (i.e., decays as fast as it is replenished by nova outbursts), the total mass of ²⁶ Al, , presently existing in the Galaxy that is produced by ONeMg novae is the sum of for all systems in the population containing an ONeMg WD, multiplied by the mean lifetime,  yr, of ²⁶ Al:

We note that in Eq. (9) is independent of the nova outburst frequency. The mass ejected within a time, , is determined by the mean mass transfer rate during ; it doesn't matter if the outburst frequency is high with a correspondingly small ejected mass per outburst, or low with a large envelope mass. This is important since, as a result, our prediction for the ²⁶ Al production by ONeMg novae is not affected by either the uncertainty in the predicted ejecta masses from nova models or by the uncertainty in the ignition condition in TNR models for nova outbursts which we discuss below.

Whenever results from population synthesis calculations similar to the one used here are given as absolute numbers, they appear either as the local mid-plane space density, , the surface density, (the space density integrated in the z -direction, i.e., perpendicular to the Galactic plane), or as a total number of systems in the Galaxy. To prevent any confusion, we note that our models are normalized either in terms of n or , and we convert n into by assuming an exponential drop-off in z direction with a scale height  pc, i.e.  pc. The total number of systems in the Galaxy is then , where A is the area of the Galactic disk; we choose  kpc² (Ratnatunga & van den Bergh 1989). We emphasize that these normalizations matter only if one is interested in absolute numbers, and authors preferring a different normalization or conversion may simply renormalize our results accordingly.

3.2. From CVs to novae

The CV population models discussed above can be used to predict the distribution of the nova outburst frequency

over system parameters (e.g., the orbital period), and the total Galactic nova rate given some ignition criterion determining the envelope mass at ignition. A comparison with observed collective properties of novae could, in principle, serve as an independent check of the models (Kolb 1995a). However, the sample of novae with determined orbital periods is small (e.g. Ritter & Kolb 1995), and observational selection effects may influence the distribution considerably.

As we shall see later, for our purposes in this paper, it is nevertheless interesting to determine the Galactic nova rate predicted by our CV population models. We do this according to three different prescriptions for : First, we adopt the "classical" criterion according to which the TNR ignites when the pressure at the base of the accreted envelope surpasses a critical value  dyn cm^-2 (Truran & Livio 1986). This criterion gives

Second, we use an approximation for the actual ignition masses based on the models of Politano et al. (1995) for the case of accretion onto massive WDs,

We note that this fit to their data is strictly valid only for . Finally, we also test the predicted nova rate by applying results of an extended grid of nova models by Prialnik & Kovetz (1995). Following Kolb (1995a), we approximate their tabulated values of ignition masses for cold WDs ( K) by , where is the canonical value for given by Eq. (11), and

(typically ). We emphasize that the computations by Politano et al. (1995, 1996) do not form a consistent set with TNR models by Prialnik & Kovetz, which rely on different assumptions. In particular the latter adopt a diffusion-convection mechanism to determine the degree of mixing between envelope and WD material, and the criterion for mass ejected from the system is different.

Integrating the distribution of over the total present CV population yields the predicted total Galactic nova rate . Table 2 lists the nova rate we obtain for the population models discussed in Sect. 4. We note that the observed value for the Galactic nova rate is subject to considerable uncertainties and the quoted value varies from author to author (see e.g. the discussion in Della Valle & Duerbeck 1993). The major uncertainty is the unknown fraction of outbursts that are missed due to interstellar (or intergalactic) absorption. In comparing our population models to observations, we use the most recent value for the Galactic nova rate,  yr^-1, obtained by Della Valle & Livio (1994). They extrapolated the observed nova frequency in galaxies of different Hubble type to the Milky Way.

Table 2. Predicted Galactic nova rate (in yr^-1) for models of set A (high correlation of component masses in ZAMS progenitor binaries) and set B (no correlation). , and were obtained using Eqs. (11), (12) and (13) as ignition criterion, respectively. The column entitled "model" gives an internal model number.

In contrast to the total nova rate, the fraction of novae which occur on a ONeMg WD,

is independent of the normalization chosen for the population model. Since ONeMg novae contain high-mass WDs, the quantity is very sensitive to the ignition criterion close to the Chandrasekhar mass. Both the simple analytic form (11) and the approximate correction factor (13) are no longer valid in that regime. Rather, to compute we use (12) as the actual result of detailed models of a TNR on ONeMg WDs for high-mass WDs, and (11) for low-mass WDs. We change from (11) to (12) at , where both criteria give the same ignition mass. An added complexity is that not all outbursts which occur on ONeMg WDs may result in an ONeMg novae. It is believed thermonuclear-powered recurrent novae must of necessity occur on WDs with masses very close to the Chandrasekhar limit because of their very short recurrence times, yet the ejecta in these systems show no evidence of enrichment in heavy elements (e.g., Starrfield et al. 1985; Webbink et al. 1987; Selvelli et al. 1992). The small number of well-studied novae with reliable abundance determinations leaves the "observed" value for also uncertain (cf. the discussion in Livio & Truran 1994). In the recent compilation by Starrfield et al. (1996), 5 out of 19 well-studied novae have neon abundances solar and may safely be considered as ONeMg novae. This sample, to which Nova Her 1991 (another ONeMg nova, e.g., Matheson et al. 1993; Vanlandingham et al. 1996) may now be added, suggests that of all observed novae occur on ONeMg WDs. This fraction could be even higher if those systems with "marginal" neon abundances are included (i.e., marginal in the sense that the observed neon abundance is only solar so that the presence of an underlying ONeMg WD cannot be unambiguously deduced; e.g., Livio & Truran 1994). Additional problems arise from selection effects not taken into account by our models; see the discussion in Sect. 5.2.3.

3.3. Uncertainties

Before proceeding to our results we wish to emphasize that the largest uncertainties in computing according to (9) come from two quantities:

First, the detailed WD mass distribution for CVs containing ONeMg WDs is unknown. In particular, for WDs formed in binaries, there is no agreement on the critical WD mass, , separating the less massive carbon-oxygen (CO) WDs from the more massive ONeMg WDs. An often-quoted value is (e.g., Iben & Tutukov 1985; Nomoto & Hashimoto 1987).

This uncertainty in arises chiefly because of the lack of detailed model calculations which follow the evolution of intermediate-mass stars beginning when they have hydrogen-rich envelopes through to the formation of an ONeMg WD. All studies begin with the star already in the core-helium burning phase. Indeed, the strongest inference that ONeMg white dwarfs can be formed does not come from theoretical models, but from the observed, large enrichments of intermediate-mass elements in neon novae. We, therefore, have approximated the WD mass spectrum of ONeMg WDs in newly-formed CVs with the WD mass spectrum of CO WDs in newly-formed CVs for the appropriate mass range. If the formation rate of CVs containing ONeMg white dwarfs and its associated distribution over WD mass is significantly different than that for CVs with high-mass CO white dwarfs, then this could have a significant effect on our predicted values for ²⁶ Al production from ONeMg novae. A more detailed consideration of the formation of CVs with ONeMg WDs is currently in progress (Politano 1996b). In our calculations, we deal with the uncertainty in by showing all results as a function of this parameter.

Second, the degree of mixing between the accreted companion material and the underlying WD material, quantified by the parameter in Eq. (2), is essentially unknown. Although it appears clear that the observed abundance of heavy elements in nova envelopes requires some mixing to take place, the physical mechanism responsible for this mixing - diffusion, shear mixing and convective mixing have been suggested - is poorly understood. The degree of mixing plays a major role for determining the ²⁶ Al yields since, for (i.e. no enrichment and thus no seed nuclei for ²⁶ Al), ²⁶ Al is produced in only negligible amounts (Hillebrandt & Thielemann 1982; Wiescher et al. 1986; Weiss & Truran 1990). Again, we treat as a free parameter and show the results as a function of . We further assume that is the same for the entire population.

© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998
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