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

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6. Results

6.1. The parent distribution function

Equation (18) was evaluated to derive the cumulative parent distribution function [FORMULA] of nova progenitors from the observed distribution [FORMULA] under the quoted assumptions and approximations. It was transformed into the usual (non-cumulative) distribution function by numerical differentiation after having been smoothed by a third order spline fit. This smoothing procedure is required to remove spurious structures caused by accidental groupings of data in the observed distribution. However, it will also remove any real structure on time scales below about an hour. This is the price to be payed for using the cumulative distribution to avoid the uncertainties due to arbitrary binning. The parent distribution is shown on a logarithmic scale Fig. 3 as a solid curve. It was calculated with the parameters listed in Table 2 and normalized in such a way that its integral over all periods is equal to the observed number of nova periods. We will refer to this distribution as the "standard case". The vertical bars indicate the actually observed nova periods.

[FIGURE] Fig. 3. Parent distribution function [FORMULA] of classical novae (solid line) calculated with the parameters listed in Table 2 (standard case) together with error margins as derived from bootstrap replications (broken line). The vertical bars indicate observed periods.


Table 2. Parameters used for the "standard case"

The observed distribution is based on only 28 novae. It will therefore be sensitive to errors due to small number statistics. In order to assign error margins to the parent distribution due to this effect, a bootstrap method was applied: Taking the observed distribution as representative, 1000 times 28 periods were chosen at random from it. From each of the 1000 resulting distributions the parent distribution was calculated. At each period the "standard deviation" was derived (note that neighbouring periods are not statistically independent and therefore the formally calculated standard deviation must be interpreted with caution) after a [FORMULA] clipping with [FORMULA] in order to reject outliers mainly caused by oscillations of the spline fit in the (unavoidable) cases of some queer accidental period distributions drawn from the observed one by the bootstrap algorithm. The range defined by these standard deviations is limited by the broken lines in Fig. 3.

The parent distribution declines from a maximum at short periods almost exponentially (as indicated by the overall linear trend in [FORMULA]) with the period. Intrinsically, there are of the order of [FORMULA] times more novae with periods [FORMULA] than with [FORMULA], although the observed distribution is about the same at both periods (Fig. 2).

In the first place the shape of the [FORMULA] distribution is due to the strong dependence of [FORMULA] on P: A much longer time is required for short period systems with a low mass transfer rate until enough mass is transferred to permit a nova outburst. The observed number of outbursts in short period systems being of the same order as that in systems with long periods, the intrinsic number of the former must be drastically higher.

The effect of the [FORMULA] relation on the parent distribution is enhanced by the influence of the ratio [FORMULA] under the integral in Eq. (18). This can be seen as follows: If P attains a large value, [FORMULA] is also large due to the well-known [FORMULA] relation. Then, only high mass white dwarfs contribute to the integrals in Eq. (18) because of the dependence of the integration limits on [FORMULA] via Eqs. (19) and (20). Since high mass white dwarfs have a small radius (which enters with the fourth power here!) the integral mean of [FORMULA] will increase with P, and - since it appears in the denominator of Eq. (18) - leads thus to smaller intrinsic numbers of systems with long periods.

The strong influence of [FORMULA] on the parent distribution is visualized in Fig. 4. The solid line is again the distribution according to the standard case. The dashed and dotted lines correspond to the cases [FORMULA] and [FORMULA], respectively, i.e. to the limits of [FORMULA] as found by Prialnik et al. (1989) and Starrfield et al. (1986). The dashed-dotted line results from calculations assuming the simplified relation between [FORMULA] and [FORMULA] according to MacDonald (1984) (see Sect. 4). As expected, changing [FORMULA] and thus the influence of [FORMULA] modifies the parent distribution considerably. This may cause some concern because even if we were confident that [FORMULA] is a good approximation, modifying [FORMULA] is equivalent to modifying the exponent of [FORMULA] in the [FORMULA] relation which is subject to uncertainties. This is also indicated by the thin line resulting from calculations, where the [FORMULA] law according to Iben et al. (1992) was exchanged by Patterson's (1984) corresponding relation.

[FIGURE] Fig. 4. Parent distribution function [FORMULA] of classical novae calculated using different prescriptions for the mass transfer rate and its influence on novae recurrence times. For details, see text.

Compared with [FORMULA], the parameters [FORMULA] and [FORMULA] have a much smaller influence on the [FORMULA] distribution. It is shown in Fig. 5 where along with the standard case (solid line) the distributions according to [FORMULA] and [FORMULA] are shown as thick dashed and dotted lines, respectively. Changing the [FORMULA] dependence by increasing/decreasing [FORMULA] modifies the parent distribution in the sense predicted qualitatively above. The thin dashed and dotted lines refer to the cases [FORMULA] and [FORMULA]. All graphs remain well within the statistical uncertainties of the standard distribution. Thus, the choice of [FORMULA] and [FORMULA] is not critical.

[FIGURE] Fig. 5. Parent distribution function [FORMULA] of classical novae calculated using different values of the [FORMULA] exponent [FORMULA] and the technical selection function exponent [FORMULA].

6.2. Comparison with population synthesis calculations

The parent distribution derived here can be compared to the distributions derived theoretically from population synthesis calculations. Kolb (1993) presented encompassing calculations for the CV population in general. While both, his and our approaches are not completely independent - in both cases the same theoretical white dwarf mass spectrum for CVs was used - they are sufficiently different to permit a useful comparison: We used the observed distribution together with the basic theory of nova outbursts to calculate the intrinsic distribution, taking into account selection effects, while Kolb used the result of calculations of the common envelope phase and evolutionary scenarios of CVs to achieve the same goal.

While the small number of observed nova periods clearly inhibits the comparison of details of the parent distribution with structure apparent in the population synthesis calculations - rendering any decision between the different models considered by Kolb impossible - the general shape of the parent distribution is not grossly different from Kolb's distributions (with the exception of the period gap; see 6.3) in the period range encompassed by the observations. This is shown in Fig. 6, where the standard case (solid line) is shown together with the populations according to models pm1 and pm3 (dotted and dashed thin lines, respectively) of Kolb (taken from his Fig. 4). Between [FORMULA] and [FORMULA] his distributions exhibit on the mean a decrease in [FORMULA] of [FORMULA] versus 3.4 in our standard case. In view of the strong dependence of our results on the uncertain [FORMULA] relation this difference is hardly significant. However, Kolb's distributions predict an excess between [FORMULA] and [FORMULA] and a quicker drop for [FORMULA] than our distribution.

[FIGURE] Fig. 6. Comparison of the parent distribution function [FORMULA] of classical (solid line) novae with distributions derived from theoretical population synthesis calculations (taken from Kolb 1993) (dashed and dotted thin lines). The thick dashed line is the distribution calculated under the assumptions of the cold nova model. The thin vertical lines indicate the limits of the period gap as defined by Kolb. For details, see text.

In another paper Kolb (1995) used the results of his population synthesis calculations together with an expression for the accreted mass [FORMULA] required to initiate an outburst to predict the distribution of classical novae, defined through their eruptions, under various model assumptions. The resulting distributions should thus correspond to the observed one, were it not for the observational selection effect which Kolb did not consider. In this sense, his calculations are (partly) equivalent to ours, but going in the reverse direction. In fact, the solid line in his Fig. 1 is - within the period range so far observed in nova systems - not unlike the observed period distribution (disregarding the period gap) shown in Fig. 2: The density of systems at low periods is somewhat less than at intermediate ([FORMULA]) periods and drops off towards higher values. The strong peak at very small periods in Kolb's (1995) distribution and also in his populations synthesis calculations (Kolb 1993) is beyond the observed short limit. The implications of this fact will be discussed in Sect. 6.4.

Kolb (1995) used the same prescription for the accreted mass [FORMULA] required to ignite a nova outburst as we did in the standard model (i.e. [FORMULA] =0). However, he also considered a modified model, the cold nova model, where [FORMULA] depends on [FORMULA] in a different way. This results in a nova distribution with a significant deficit of outbursts at small periods and an enhancement at intermediate ones (dotted line in his Fig. 1). We calculated the parent distribution function using the same law for [FORMULA] and show it as a dashed thick line in Fig. 6. While the overall inclination at intermediate periods is not unlike that of Kolb's (1993) synthetic populations, it would predict a much too small relative number of short period novae. Thus, considering the period distribution from Kolb's population synthesis, the observed nova period distribution does not support the cold nova model.

6.3. The period gap

While we thus find an overall agreement of the intrinsic period distribution of novae and the theoretically predicted general CV distribution, the observed nova distribution and in consequence the calculated parent distribution does not show the period gap which evolutionary scenarios for CVs predict and which is so obvious in the overall observed period distribution. This lead Baptista et al. (1993) to the conclusion that no such gap exists for classical novae. From his theoretical calculations Kolb (1993) finds the period gap to be defined by the limits [FORMULA]. This is in reasonable agreement with the observed gap for the overall CV population: A period distribution function constructed from data in the Ritter catalogue (Ritter & Kolb 1993) clearly allows the empirical gap to be defined as [FORMULA]. Three novae (V Per, QU Vul and V2214 Oph) are observed to lie in this range.

Due to CVs born in the gap and young enough not yet to have evolved out of it, the gap is not expected to be totally devoid of novae. The calculations of the relative number of nova events by Kolb (1995), based on his synthetic CV population models (including the gap) can be used to find the probability to observe a given number of systems in the gap. Since Kolb disregarded observational selection effects a comparison with the observed distribution is not strictly possible. However, the overall similarity of the observed and calculated distribution (see Sect. 6.2) secures that the error will remain moderate.

Taking Kolb's (1995) distribution and the limits of the period gap (his Fig. 1, solid line) which is only marginally smaller than found in his previous study, the probability that out of a randomly chosen sample of 28 periods exactly 3 (3 or more) fall into the gap is only 1.1% (1.3%). So small probabilities justify concern and the question if classical novae have a way to avoid detaching and thus to stop mass transfer when passing through the classical period gap.

6.4. The low-period cutoff

Kolb's (1993) population synthesis calculations predict of the order of two times more CVs below the observed low period limit of CVs in general and classical novae in particular than above it. The predicted minimum period lying below the observed one can be explained by the insufficient knowledge of stellar opacities at low temperatures, rendering the calculated period cutoff highly uncertain. This issue is discussed by Kolb (1993) as well as by Rappaport et al. (1982).

Thus, the predicted presence of a large number of CVs below the observed period limit may just be due to an uncomplete understanding of CV evolution. However, this does not resolve the problem but only shifts it: These systems should then populate the range just above the empirical minimum period and thus the number of novae should be raised above the observed number. Therefore, compared to the overall synthetic CV population, classical novae appear to be deficient in the range between minimum period and period gap. However, as a caveat it must be mentioned, that on the observational side we are always dealing with small number statistics.

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© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998