4 Time-series photometry

The observations were done at the Nordic Optical Telescope (NOT) with the Andalucia Faint Object Spectrograph and Camera (ALFOSC), equipped with a Loral, Lesser thinned, $2048\times2048$ CCD chip, and modified with our own control software to be able to observe in high-speed multi-windowing mode.

The sky area available for locating a reference star is limited to ${\sim} 6.5\times6.5$ arcmin².

Figure 4: Amplitude spectra for HS0039+4302. The three upper panels show the individual spectra for the three nights, and the lower panel shows the spectrum of the three consecutive nights taken together. The two inset panels represent a section of the same spectrum with a larger scale (left) and its spectral window (right).

4.1 Observations and reductions

Table 4 contains the information related to the time-series observations. We observed the stars in windowed mode using three or more reference stars for constructing the relative light curve. All the observations were made with a standard Bessell B-band filter (NOT #74, Bessell 1990), except for the last observation run of HS0444+0458, where we experimented with a much wider filter that effectively encompass all bands from B to R with more than 90% transmission (NOT #92). This filter (hereafter referred to as the W filter) has the same center as that of the V-band (5500Å), but is a full 2750Å wide. The cycle time was set to 20s, except for the first run of HS0444+0458, where the cycle time was set to 30s. For the 20s cycles, the actual exposure time varied between 15.6 and 16.4 s, depending on the number of reference stars selected (3 or 4) and the sizes of the readout windows ( $42\times42$ to $52\times52$ pixels).

The data were reduced on-line using the Real Time Photometry (RTP) program developed by one of us (R. Østensen) as part of his Ph.D-project (Østensen 2000). Some details about this software are given in our previous paper (Østensen et al. 2001).

The processing includes bias level removal, flat fielding, sky subtraction, extinction correction and aperture photometry using optimal apertures that track each stars geometrical center. The resulting light curves are shown in Figs. 3, 5, 7 and 9, and are discussed below for each of the four stars. The light curves show differential photometry between the target and the best combination of the available reference stars.

Figure 5: Observed and synthetic light curves for HS0444+0408. Note that since the observations on the second night were done 2 hours later in the night, the panel has been shifted by 0.1 JD relative to the bottom scale.

The optimal aperture for each run was selected after processing all data sets with apertures of a wide range of diameters and choosing the one that gave the best signal-to-noise ratio in the Fourier Transform (FT), using the amplitude of the primary peak for the signal and the mean of the amplitude spectrum outside the pulsation range for the noise. The apertures tested ranged between 10 and 40 pixels, which corresponds to 1.9 and 7.6 arcsec on the sky, while the best results stayed within the range 18 to 22 pixels (3.4 to 4.2 arcsec).

Figures 4, 6, 8 and 10 show the FTs of the light curves. The significant peaks detected in these amplitude spectra are listed in Table 3.

4.2 Data analysis and results

4.2.1 HS0039+4302

Since the runs from the three nights have progressively longer time spans, the corresponding amplitude spectra reveal increasing detail. The dominant period is seen at 5.14 mHz, with an amplitude of about 8 mma. In the amplitude spectrum of the third night, four significant peaks are revealed (see Table 3 for details). In the bottom panel of Fig. 4, the combined FT of all the three nights taken together is shown. This spectrum agrees with what is seen in the third night alone, but reveals no further significant peaks.

Figure 3 shows the observed data superimposed on a synthetic light curve computed from the four detected pulsation periods. The match between these are quite good, although the amplitude spectra show some remaining power close to the observed oscillation frequencies (Fig. 4). The peak at 4.6mHz in the amplitude spectrum of the first run vanishes completely after prewhitening.

The average noise level in the amplitude spectra after prewhitening of the four detected periods is below 0.5 mma for the first run, below 0.3 mma for the second, and about 0.35 in the third run. The average noise level in the combined spectrum is below 0.25 mma.

4.2.2 HS0444+0408

Figure 6: Amplitude spectra for HS0444+0408. The three upper panels show the individual spectra for the three nights, and the lower panel shows the spectrum of the three consecutive nights taken together. The two inset panels represent the same spectrum with a larger scale (left) and its spectral window (right).

In the first run on HS0444+0408, a sequence interval of 30s was used. After the discovery of the clear pulsation on the first night, we decreased the sampling interval to 20s for the second and third runs, to better sample the light curve. On the third night we observed this target using the W-band filter, in an attempt to improve the signal-to-noise ratio. In this way the white noise in the FT dropped from 0.5 mma in the second run to 0.35 in the third, even when the run length in the third run was slightly shorter than in the second.

Both the significant peaks detected, the dominating one at 7.31 mHz and the secondary one at 5.86 mHz, have a quite stable amplitude over the three nights of 11-12 and 2.3-2.5 mma respectively. In particular, the amplitudes of both peaks in the third night, when we used the W filter, are almost identical to the amplitudes obtained in the first two nights, when the B filter was used. Therefore we could do a combined FT using the data from all the three nights together.

The synthetic light curve based on the two detected periods, and plotted with a dotted line in Fig. 5, reproduces the observed light curve very well.

In the bottom panel of Fig. 6, the combined FT of the three nights is shown. The feature between the two main periods is just above the detection limit, at 1.4 mma. The average noise level in the combined spectrum is below 0.3 mma.

The extinction corrected light curves indicate an average magnitude B = 15.4for both nights with B-band data, in good agreement with the estimates based on the photographic plates.

4.2.3 HS1824+5745

Figure 7: Observed and synthetic light curves for HS1824+5745.

Figure 8: Amplitude spectra for HS1824+5745. The three upper panels show the individual spectra for the three nights, and the lower panel shows the spectrum of the three consecutive nights taken together. The two inset panels represent the same spectrum with a larger scale (left) and its spectral window (right).

This target and the following show pulsation amplitudes of less than half of the previous two. For this reason the light curves and amplitude spectra have been scaled to show as much detail as possible.

The Fourier transforms of HS1824+5745 show no indications of more than one single period at 7.21 mHz. However, the synthetic light curve produced from the detected period does not reproduce the observed light curve as well as what was found for the two previous sdB pulsators (see Fig. 7). The noise in the data is higher in these observations, due to a combination of poorer weather conditions and a fainter target. Note that since the pulsations are much weaker, the y-axis scale in Fig. 7 has been stretched relative to the light curves in Figs. 3 and 5. Note also that the last third of the first night contains data of poorer quality than the first part, due to an onset of clouds that eventually forced the termination of observations. The second night has some slight cirrus activity throughout the run, while the third night is photometric.

The amplitude appears to be variable, changing from 2.6 to 4.8 mma from the first to the third night, but this is uncertain due to the poor conditions on the first two nights and the short length of the runs. For instance, the amplitude for the first night can be brought in accordance with that of the whole run by taking the FT of only the first (good) two thirds of the sequence.

4.2.4 HS2151+0857

Figure 9: Observed and synthetic light curves for HS2151+0857.

Figure 10: Individual and combined amplitude spectra for HS2151+0857.

This object shows a richer period spectrum than HS1824+5745, and clearly reveals four periods at 6.59, 6.86, 7.43 and 7.72 mHz (see Fig. 10), with amplitudes between 2 and 5 mma, as listed in Table 3.

After prewhitening of the four tabulated frequencies, the only significant feature in the combined amplitude spectrum is a 1.4 mma peak at 7.72 mHz, only 6 $\mu$Hz away from the 3.3 mma peak. This feature may not be real, since the amplitude of this peak in the individual spectra appears to change considerably from the first to the third night. The synthetic light curve trace the obseved data reasonably well, but some discrepancies are seen. Some of this is due to noise, since this target is about one magnitude fainter than the three others, but it is also likely that unresolved periods or changes in period amplitudes over the three nights are significant.

The mean noise level in the prewhitened spectra is below 0.8 mma in the first night, below 0.7 mma in the second, and below 0.6 mma in the third. In the prewhitened spectrum of all three nights together the mean is below 0.4.

The prescision of the observations decreases somewhat in the last quarter of the third run, due to the onset of twilight.