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Astron. Astrophys. 322, 807-816 (1997)

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4. The outburst frequency of novae

Obviously, at a given orbital period the number of observed novae should increase in proportion to the total number of outbursts occuring in the nova systems. Thus, the observed distribution of orbital periods [FORMULA] may be written as a function of the intrinsic number of potential novae with this period [FORMULA] and corrected for the selection effects discussed in Sect. 1:

[EQUATION]

Here, [FORMULA] is the mean time between nova outbursts in a sample of novae with known periods, [FORMULA] is the average theoretical recurrence frequency corrected for the selection effect [FORMULA]. [FORMULA] is the technical selection function. If we assume that the average mass transfer rate is a function of the orbital period and equal to the mass accretion rate [FORMULA], the theoretical recurrence time of a system may be expressed as a function of the orbital period:

[EQUATION]

where [FORMULA] is the mass accreted between outbursts which is a function of the white dwarf mass, of the mass accretion rate itself, of the preoutburst white dwarf luminosity [FORMULA] and of the metallicity of the core-envelope interface [FORMULA] (Starrfield 1989). The mass transfer rate in nova precursors as a function of the orbital period is not well known. We shall adopt here the estimate of Iben et al. (1992), i.e. [FORMULA] = [FORMULA] for systems subject to magnetic stellar winds (systems above the period gap), where [FORMULA] is a parameter of order unity, and [FORMULA] = [FORMULA] for systems only subject to gravitational wave radiation (systems below the period gap). Additionally, the empirical [FORMULA] relation of Patterson (1984) will be used for comparison. Using the secondary mass-period relation in the form given by Patterson (1984), [FORMULA] becomes thus a function of P. One should be aware, however, that this [FORMULA] relation is accurate only to the order of magnitude. But note that in the present context only the functional dependence of [FORMULA] on [FORMULA] and thus P is of importance, not the absolute value. Following Fujimoto (1982a) we may express [FORMULA] in terms of a critical pressure [FORMULA], for detonation. According to the results of TNR models (e.g. Livio 1988, and Starrfield 1989) and analytical calculations (MacDonald 1983) the white dwarf luminosity has a weak dependence on the critical envelope mass. Neglecting thus the dependence of [FORMULA] on [FORMULA] can be written as a simple force-pressure relation:

[EQUATION]

where [FORMULA] is the white dwarf radius. Neither the numerical value of [FORMULA] nor its functional dependence on [FORMULA] and [FORMULA] are well known. Thus, the same holds true for the critical mass [FORMULA]. We adopt here the ansatz of Ritter et al. (1991), assuming that [FORMULA] can be written as:

[EQUATION]

In principle, the parameters [FORMULA] and [FORMULA] are unknown. However, the ansatz is only sensible if [FORMULA] is close to 1 and [FORMULA] close to 0, implying that the dependence on the white dwarf mass is mainly determined by its effect on the hydrostatic pressure, and that the dependence on [FORMULA] is small. Semi-analytical computations by MacDonald (1984) yield [FORMULA]. Even lower values ([FORMULA]) were found in the models of Prialnik et al. (1989) and Starrfield et al. (1986). Concerning the [FORMULA] -parameter, model values for the accreted mass at ignition as a function of the white dwarf mass (Fujimoto 1982b and MacDonald 1984) indicate [FORMULA]. We therefore consider [FORMULA] and [FORMULA] as a reasonable first approximation. However, the effect of deviations from these values will be explored. In particular, we will permit [FORMULA] to depend on [FORMULA] as suggested by the semi-analytical calculations of MacDonald (1984). We approximate his [FORMULA] relation by a linear fit to the values in Table 1 of Ritter et al. (1991). The dependence of [FORMULA] on chemical composition will be disregarded because there is no reason to assume a relation between the metallicity of the CEI and the orbital period. A possible relation between the mass of the white dwarf (the distribution of which might depend on the orbital period for evolutionary reasons; but see Appendix A) and CEI metallicity will be neglected in the first approximation. Thus, the unknown numerical value of [FORMULA] becomes of no consequence in the present connection since, as a constant, it influences the absolute number of outbursts at a given period but not the period distribution. Combining Eqs. (13) and (15), we write:

[EQUATION]

This relation defines the theoretical recurrence time for a single system with a given orbital period and white dwarf mass. In the next step we will derive the observed recurrence time of the sample of systems as a function of the parent period frequency distribution [FORMULA]. The white dwarf mass distribution for every period bin will be allowed for by computing the average recurrence frequency over the theoretical mass spectrum of CVs (Politano & Webbink 1990, Kolb 1993). This results in a weighted average frequency of outbursts in an ensemble of systems with a given orbital period. The white dwarf radius may be calculated using the Hamada & Salpeter (1961) relation for cold carbon white dwarfs. In the following treatment it is implicitely assumed that the recurrence time as given by Eq. (13) (typically [FORMULA] to [FORMULA] years) is much shorter than the time scale for binary evolution. Under these assumptions it is

[EQUATION]

and we can solve Eq. (12) for [FORMULA] using Eqs. (11) and (16):

[EQUATION]

The upper limit of the integrals corresponds to the Chandrasekhar mass while the lower limit is set to count only the dynamically and thermally stable systems, defined by the critical mass ratio:

[EQUATION]

[FORMULA] is the secondary mass as a function of the period. The calibration of Patterson (1984) was used as before. The value of [FORMULA] depends on the structure of the secondary star. The values of Politano & Webbink (1990) are adopted:

[EQUATION]

For computing the integral mean in Eq. (18), a first approximation to the surface density of CVs is required. It is assumed that the white dwarf mass distribution integrated over the orbital period at present is close to the ZACB value. This approximation is justified since most of the accreted mass is ejected in the outbursts. To a good approximation the white dwarf mass can be considered as constant on long time scales, the mass transferred from the secondary between outbursts being removed by the eruption. The secondary mass diminishes as the system evolves by losing angular momentum. Conservative mass-transfer between outbursts [FORMULA] will be assumed. At a time [FORMULA] after the birth of the CV, the secondary mass is simply given by

[EQUATION]

where [FORMULA] is the mean mass transfer rate. Therefore, the distribution of zero age masses of Politano & Webbink (1990) should be corrected for evolutionary effects. The approximate treatment of the secondary secular evolution proposed by Ritter et al. (1991) will be adopted here. Since we use the distribution [FORMULA] instead of [FORMULA] the transformation

[EQUATION]

is applied where

[EQUATION]

is the Jacobian of the transformation between [FORMULA] and [FORMULA] coordinates. The second equality in Eq. (23) is understood considering the [FORMULA] - P - relation of Patterson (1984) and the fact that [FORMULA] does not depend on P. Hence, the resulting Jacobian determinant is only a function of P and cancels out on the right side of Eq. (18).

To solve Eq. (18) the function [FORMULA] must be specified. This can be done by combining the observed periods in bins of arbitrary widths and centres. However, the resulting parent distribution will then depend on the chosen bin parameters. To remain independent of this effect it is preferable to calculate the cumulative distribution function, i.e. the number of novae with a period above a given period P, as a function of P. This is achieved by integrating both sides of Eq. (18) from infinity down to P. Due to the discrete values of the observed nova periods, the integral over the right hand side of Eq. (18) decays into a sum of terms, each corresponding to an observed period. The cumulative parent distribution then becomes a step function with steps at the observed nova periods.

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© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998

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