## 3. Nova population modelsIn addition to knowledge of how much 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 ## 3.1. Model assumptions for CV populationsWe 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 Of the parameters which enter into a calculation of the formation
rate of CVs, the initial distribution of the mass ratio, 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 Assuming that We note that in Eq. (9) is
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
## 3.2. From CVs to novaeThe 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 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
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
## 3.3. UncertaintiesBefore 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 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
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