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Astron. Astrophys. 327, 173-182 (1997)

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3. Data analysis

3.1. Photometry

3.1.1. Period analysis

The characteristic eclipse features of HS 1804 + 6753 are displayed in Fig. 7, which are applied to derive the orbital period. The phase points [FORMULA] and [FORMULA] indicate ingress and egress of the hot spot, [FORMULA] and [FORMULA] that of the white dwarf. The most stable well-defined eclipse feature is the mid egress of the white dwarf. During the eclipse ingress additional light from the hot spot may disturb the light curve. The exact mid egress time is determined by fitting an approximating spline to each of the light curves and by calculating their derivatives. At the mid egress of the white dwarf, the derivative shows a local maximum, which gives the mid egress time with an accuracy of 4 s. Assuming that the orbital period is constant over the observation period, we derived its value by fitting a straight line through these points. Since we know the duration of the eclipse of the white dwarf (see Sect. 3.1.2), the ephemeris of its mid eclipse time [FORMULA] is calculated to


The measured eclipse timings and their corresponding cycle numbers are given in Table 5 and 6. However, from a plot showing residual vs cycle-number (Fig. 8), the orbital period does not seem to be constant. This may be caused by some unknown distortion of the eclipse egress or by a shift of this feature indicating real period changes. In this case, possible explanations include influence by a third body in the system or, more likely, magnetic activity of the secondary star (Warner, 1988). Similar period fluctuations on time scales of years have been observed e.g. for IP Peg (Wolf et al., 1993).

[FIGURE] Fig. 7. Mean B light curve (crosses) and its corresponding derivative (solid line). The mid ingress and mid egress times of the hot spot ([FORMULA]) and of the white dwarf ([FORMULA]) are marked.

[FIGURE] Fig. 8. O-C residuals of the eclipse egress times of the white dwarf calculated from the ephemeris given in Eq.(1).

3.1.2. Eclipse times

In order to separate the eclipse of the hot spot from that of the white dwarf a mean light curve was derived from the data using the ephemeris given in Eq. (1). The mean B light curve together with its derivative are plotted in Fig. 7. Due to the temperatures of both the hot spot and the white dwarf and due to the [FORMULA] -dependent quantum efficiency of the detectors, the B channel provides a maximum signal-to-noise ratio. The local extrema of the derivatives taken from the B channel give the four phase points [FORMULA]


[FORMULA] defines the mid egress of the white dwarf, while [FORMULA] defines that of the hot spot. The small error bars are due to superposition of 50 light curves, assuming again the orbital period to be constant. The duration of the white dwarf's egress can not be measured due to considerable radiation from the disk. Nevertheless, this does not affect the determination of the mid egress.

In principle, in order to assign the ingress phase points [FORMULA] and [FORMULA] to the respective components, two methods can be applied. During an outburst most of the optical radiation originates from the disk. Assuming a symmetrical disk, its eclipse should be symmetrical to the mid eclipse of the white dwarf. Thus, from the observed mid eclipse time during an outburst and from the observed mid egress time during quiescence, the mid-ingress time of the white dwarf can be calculated. However, this method can not be applied to HS 1804 + 6753 since the eclipse light curve is asymmetric in outburst (see Fig. 6). On the other hand, an ingress/egress phase diagram can be used together with a single particle trajectory (SPT) to determine the position of the hot spot. As shown in Sect. 3.1.3 a unique solution can be found if the hot spot is eclipsed first. Therefore, the duration between mid ingress and mid egress of the hot spot [FORMULA] and that of the white dwarf [FORMULA] is


with a time difference [FORMULA] of the mid eclipses


3.1.3. Photometric mass ratio

The mass ratio [FORMULA] of the components of HS 1804 + 6753 was calculated from their respective radial velocities. The value of the mass ratio is confirmed by solving the SPT under the influence of the Roche potential (Dreier, 1986). This photometric method yields solutions for the SPT assuming that the hot spot is eclipsed first. Different SPTs fitting the equations yield mass ratios between 0.7 and 0.8 in close correspondence with the spectroscopic results (Fig. 9).

[FIGURE] Fig. 9. Different trajectories for mass ratios 0.5... 1.0. The hot spot is identified by the dot at (-0.55131/0.097803).

3.2. Spectroscopy

3.2.1. Radial velocity of the white dwarf

The radial velocity of the white dwarf is derived from spectra observed during quiescence. For the determination of the velocity [FORMULA] the double-peaked emission lines of [FORMULA], [FORMULA], and [FORMULA] were taken. Radial velocities derived from other emission lines turned out to be less reliable. Line profiles were measured applying the method of Schneider and Young (1980). This method employs two Gaussian bandpasses, specified by their FWHM [FORMULA] and separation a, to measure the wings of the emission line, finding the velocity shift required to make the line flux equal in the two bandpasses. This yields different [FORMULA] values as a function of these two parameters. The calculated velocities can be fitted by the equation



The value with a minimal relative error, [FORMULA] is selected (Fig. 10). From this, the weighted mean radial velocity for the white dwarf is derived to


Note that possible systematic errors have not been considered.

[FIGURE] Fig. 10. [FORMULA] velocities vs bandpass separation and their respective system velocities [FORMULA] (upper panel). [FORMULA] is marked with a [FORMULA], [FORMULA] with a [FORMULA], and [FORMULA] with a [FORMULA]. In the middle panel their respective angles [FORMULA] are plotted, and in the lower panel the minimal relative error [FORMULA] is shown.

3.2.2. Radial velocity of the secondary

The sole spectral signatures of the secondary are absorption lines of Ca I. In order to derive their radial velocities, Gaussian profiles were fitted. The value of [FORMULA] was calculated from


The resulting phase shift is nearly zero. This is consistent with the assumption that the photocentre of the secondary coincides with its mass centre. The weighted radial velocity obtained from 266 line profiles (Fig. 11) is


Note again that systematic errors have not been considered.

[FIGURE] Fig. 11. Radial velocity curve of the secondary component measured from the following absorption lines: Ca I 6103Å ([FORMULA]), Ca I 6122Å ([FORMULA]), Ca I 6162Å ([FORMULA]), Ca I 6439Å ([FORMULA]), Ca I 6450Å ([FORMULA]), Ca I 6463Å ([FORMULA]) [FORMULA]. For clarity error-bars have not been plotted.

3.2.3. System velocity

The system velocity is derived from the radial velocity curves of the two components. A sine-fit through both velocity curves yields


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© European Southern Observatory (ESO) 1997

Online publication: April 8, 1998