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Astron. Astrophys. 327, 231-239 (1997)

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3. Parameters for WW Cet

3.1. The state of WW Cet during the observations

In Fig. 2, we present averaged spectra for each of the three of our observing periods. The typical properties of the lines are summarized in Table 2.

[FIGURE] Fig. 2. Averaged normalized ITS spectra of WW Cet. These spectra represent 3 different states, as derived from the continuum flux (see Sect. 3.1): 1993-quiescence, 1994a-outburst, 1994b-decline from outburst. For clarity, offsets of 0.5 and 1.0 units have been applied to separate the spectra


Table 2. Properties of the emission lines in the mean spectra

As stated in Sect. 2.1, the flux calibration proved to be very uncertain. However, it is still considered to be good enough to provide some information about the state of WW Cet at the time of our observations. For this reason, we have measured the continuum flux close to the H [FORMULA] line. The first of the data points was taken as reference, while the other values were transformed into magnitudes. The results are plotted in Fig. 3.

[FIGURE] Fig. 3. Continuum flux of WW Cet around the H [FORMULA] line, calculated relatively to the first value of our data set ([FORMULA]). For the different nights, x is 271 (93/10/10, [FORMULA]), 272 (93/10/11, [FORMULA]), 638 (94/10/12, [FORMULA]), and 639 (94/10/13, [FORMULA])

These magnitudes show a scatter which amounts within each single data set to [FORMULA].8 in 1993 and [FORMULA].5 in 1994. However, the differences between the 1993 and the 1994 magnitudes are much larger than this, with the 1994 data being much brighter. Furthermore, the second night in 1994 shows WW Cet on a steady decline from the constant level maintained during the first night, almost down to the brightness observed in the 1993 data.

We conclude that WW Cet was observed in quiescence in October 1993, and in outburst as well as in decline from outburst in October 1994. This is also supported by the observations of the Variable Star Section of the Royal Astronomical Society of New Zealand: our observations in 1993 lie between two registered outburst maxima on JD 2 449 264 and JD 2 449 276 (Jones et al. 1993), and 1 and 2 days after the outburst maximum (V = 11 [FORMULA].0) on JD 2 449 637 in 1994 (Cragg et al. 1994).

3.2. Radial velocities

In order to measure the Doppler shift of the observed line with respect to the rest wavelength of H [FORMULA], we applied the double Gaussian method as outlined by Shafter (1983). Here, two Gaussians with the same FWHM separated by a wavelength interval d are fitted to the line. By varying the FWHM and d, one is able to measure different parts of the line, i.e. the influence of the line center will increase with increasing FWHM and decreasing d, while narrow Gaussians at wide separation will measure the radial velocites of the line wings.

This method provides an important advantage over a single-Gaussian fit, which does not account for often observed dilution of the disk-born line profile by additional emission sources. However, their influence is irrelevant if we just want to derive the orbital period from the data. At this stage we therefore chose broad Gaussians with FWHM = 40 Å, measuring those parts of the line with good S/N.

In order to derive the orbital period from these data, we applied both the phase-dispersion minimization method of Stellingwerf (1978) as included in IRAF, and the analysis-of-variance (AOV) algorithm by Schwarzenberg-Czerny (1989) as implemented in the time-series-analysis context of MIDAS2 . While the extracted periods did not differ significantly, the latter method proved more powerful in identifying alias periods.

In earlier radial velocity studies (Kraft & Luyten 1965, Thorstensen & Freed 1985, and Hawkins et al. 1990); these will be referred to hereafter as KL65, TF85, and HSJ90, resp.), no unambiguous decision could be made between two periods around 0.150d and 0.176d. Such alias problem, however, has been solved recently by Ringwald et al. (1996) (hereafter R96) in favour of the longer period. We therefore refrain from showing the actual periodograms of our study, but only report here our own results, which are


for the 1993 and 1994 data, respectively. The rather poor precision is probably due to the short overall coverage per orbit (only on 94/10/13 we observed the system for [FORMULA]).

While the broad Gaussians are sufficient to derive the orbital period, greater care has to be taken when selecting the radial velocities which are used to determine the orbital parameters, i.e. the parameters of the function


As the motion of the WD should be best represented by the variation of the line wings, we used narrow Gaussians with FWHM = 4 Å, which gave the best compromise between high resolution and high S/N for our data. To determine these parameters we used only the quiescence data. The H [FORMULA] line in the outburst data is too narrow for the line wings to be resolved, and the radial velocities of the decline data are probably disturbed by non-orbital changes within the disk (Tappert et al. 1997). The diagnostic diagram for these extracted data is shown in Fig. 4.

[FIGURE] Fig. 4. Diagnostic diagram for the 1993 quiescence radial velocity data measured with a double Gaussian with FWHM = 4 Å

In principle one can choose between two possibilities for selecting the separation which corresponds to the correct parameters from this diagram. The first is to look for the minimum value of [FORMULA] as this indicates the best fit to the data. In our case, this separation is 5 Å which is of course much too close to the line center. On the other hand, at some point the line wings will be too much affected by the noise to yield reliable parameters. This point is marked by the last [FORMULA] before the errors increase steeply. Note that this does not a priori mean that at this point one has escaped all diluting emission to obtain pure WD motion, but it is probably the best possible approximation which can be extracted from the emission lines.

For our data we have therefore chosen the parameters corresponding to a separation d = 44 Å as this provides the last [FORMULA] and the values do not deviate significantly from the ones at d = 41 Å. The corresponding parameters are


The phase shift between line center and wings is [FORMULA] = 30o, corresponding to [FORMULA] 0.08 orbits.

3.3. Towards a more precise orbital period

The publication of four previous radial velocity studies gives us the possibility to determine the orbital period with a much greater precision than before. This can be done by comparing the zero phases of a set of functions from Eq. (2) which are fitted to the single data sets for a sample of periods P within the uncertainty range but with the same [FORMULA]. Here, zero phase refers to the phase [FORMULA] when the source is in superior conjunction to the observer. In this point the radial-velocity curve fulfills


With the combined data of KL65 and our 1994 period spanning a range of 30 years (11018 days), a precision of [FORMULA] [FORMULA] days seems possible if an extrapolation with an accuracy of 0.1P is required. However, Fig. 4 shows that the zero point of the radial velocity curve strongly depends on the part of the line it was measured.

The published data of TF85 and R96 were obtained by using two broad Gaussians with FWHM = 40 Å and 45 Å, respectively. Both caught WW Cet in quiescence, and we therefore assume that our 1993 data measured with equally broad Gaussians are quite accurately in phase with these data. On the other hand, HSJ90 measured the absorption lines of the secondary which should show exactly opposite phase to the motion of the WD and thus to our 1993 wings data (HSJ90 found no significant eccentricity in their radial velocities, and the absorption lines seem thus undisturbed by irradiation effects). We therefore expect that we are allowed to compare these data with an accuracy of 0.1P.

To choose the frequency range to be searched, we took from all published periods the most accurately determined one from TF85 and allowed a deviation of 5 [FORMULA]. With a precision of [FORMULA] cycles/day this gave us a range of 5.667 970 - 5.710 044 cycles/day. We then compared the data sets in pairs by fitting functions of the form in Eq. (2) for each frequency and calculating the zero points according to Eq. (4). The resulting frequencies, i.e. those which give the same phasing (within 0.1P) for the TF85, R96, and our 1993 broad-Gaussians data on the one hand, and for the HSJ90 and our 1993 wings data on the other, are given in Table 3.


Table 3. Results of the frequency search as explained in Sect. 3.3. The upper half of the table gives the frequencies (and their corresponding periods) for the comparison of our 1993 data with TF85, HSJ90, and R96. The frequencies of the lower part were determined by including as well our 1994 data and KL65

Further alias frequencies can be removed by including our 1994 and the KL65 data. However, the phasing of the 1994 data might be somewhat different from the quiescence data (Tappert et al. 1997) and we do not know, to which part of the line the radial velocities of KL65 correspond to. On the other hand, the phases of the line center and the wings in our 1993 data differ for less than 0.1 orbits, and thus we do not expect much bigger differences between any of the data sets.

Therefore, we used for the extended comparison with the 1994 and the KL65 data included an accuracy of 0.2 P (note that the largest possible difference is 0.5 P). With this we excluded most of the aliases and are left with only 2 possible frequencies (Table 3). Of these, the period [FORMULA] seems the more probable one as it lies near the center of the searched range, while [FORMULA] is just at the edge of it, and a comparison with the periods for the single data sets in Table 4 shows that [FORMULA] is in all cases closer than [FORMULA]. The respective dates of the superior conjunction of the WD used for the extrapolations below are therefore [FORMULA] = HJD 2 449 271.597 205 7, and [FORMULA] = HJD 2 449 271.596 902 1.


Table 4. Comparison of our extracted periods with the ones of the single data sets. Not included is the (wrong) period of KL65

3.4. System parameters

With the now established period P and the semiamplitudes [FORMULA] and [FORMULA] we can - under certain assumptions - calculate further physical parameters like mass and radius of the components, and the inclination of the system. For these calculations we will use the period [FORMULA] (results obtained for [FORMULA] do not differ significantly).

Warner (1995) gives semi-empirical relations for mass and radius of the secondary,


(P is in days) for [FORMULA]. Thus we obtain for WW Cet [FORMULA] and [FORMULA]. This is in good agreement with the values for a main-sequence star of spectral type M2-3 (cf. HSJ90).

The mass of the WD can be obtained from the semiamplitudes,


Thus [FORMULA], where we used [FORMULA] = 224(7) km/s from fitting the radial velocities of HSJ90 with respect to [FORMULA]. The WD's radius is more difficult to determine, as the chemical composition is unknown, and therefore only rough estimates will be possible. Warner (1995) gives, for a non-rotating helium WD with a mass [FORMULA], the approximation


where [FORMULA] is the Chandrasekhar mass. This yields [FORMULA]. However, we estimate this number a factor 1.5 too low when compared to Fig. 1 of Hamada & Salpeter (1961). Van Amerongen et al. (1987) use


yielding [FORMULA] which seems a more reasonable result.

Finally, we can calculate the inclination i of the system. For a Roche-lobe filling secondary,


which leads to [FORMULA].

All derived parameters of WW Cet are summarized in Table 5.


Table 5. System parameters of WW Cet

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© European Southern Observatory (ESO) 1997

Online publication: April 8, 1998