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Astron. Astrophys. 350, 517-528 (1999)

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4. Time series analysis

The presence of periodicities in FO Aqr has been investigated in the FOS continua, emission lines and zero-order light as well as in the optical photometric data.

4.1. HST UV data

Fluxes in five line-free continuum bands have been measured in each FOS spectrum in the ranges [FORMULA]1265-1275, [FORMULA]1425-1450, [FORMULA]1675-1710, [FORMULA]2020-2100, [FORMULA]2410-2500. Line fluxes of He II [FORMULA]1640, N V, Si IV and C IV and [FORMULA] have been computed adopting a method which uses for the continuum a power law distribution as found from a fit in the above continuum bands. Furthermore, since the low spectral resolution of FOS data prevents the study of UV line profiles, measures of the V/R ratios of emission lines have been used to investigate possible motions in the lines. These are defined as the ratios between the integrated fluxes in the violet and red portions of the emission lines assuming as centroid wavelength that measured in the average profile. Such analysis is restricted to the strong emissions of He II, C IV and N V lines whose FWZI are [FORMULA]3000 km s-1, [FORMULA]4000 km s-1 and [FORMULA]3000 km s-1 respectively.

In order to detect the active frequencies in the power spectrum and to optimize the S/N ratio, a Fourier analysis has been performed using the DFT algorithm of Deeming (1975) on the total UV continuum (sum of the five bands) and line (sum of N V, Si IV, C IV and He II) fluxes and zero-order light. From Fig. 2, the dominance of the [FORMULA] variability is apparent, being about twice the [FORMULA] signal, as well as the presence of substantial power at the sideband and orbital harmonic frequencies. To distinguish real signals from artifacts due to the sampling of the HST orbit, a least-square technique was applied to the each data set which fits simultaneously multiple sinusoids at fixed frequencies. A synthetic light curve with the same temporal sampling of the data was created and subtracted (residuals). The continuum and zero-order light reveal, besides the [FORMULA] and [FORMULA] frequencies, also the [FORMULA], [FORMULA] and [FORMULA] sidebands, whereas in the emission lines the [FORMULA] and [FORMULA] frequencies are detected. In Fig. 2 the amplitude spectra relative to the multiple sinusoids and the residuals are also shown for comparison. It should be noted that peaks at [FORMULA] and [FORMULA] are identified as sideband fequencies of the spin and HST orbital frequencies. These are removed by the method as shown by the residuals. Then a five (four) frequency composite sinusoidal function for each spectral band (emission line) has been used and the derived amplitudes, reported in Table 2, have been compared with the average power in the DFTs of the residuals ([FORMULA]) in the range of frequencies of interest (i.e. [FORMULA] 1.4 mHz), (Column 8). While the signals at [FORMULA] (continuum) and at [FORMULA] (emission lines) on average fulfil a 4 [FORMULA] criterium, the other sidebands are between between 2.1 and 3.5 [FORMULA].

[FIGURE] Fig. 2. From top to bottom: DFTs of the UV continuum, zero-order light, emission line fluxes and V/R ratios of He II, C II and N V (solid line). DFTs of the sinusoidal functions (dotted line) together with those of the residuals (dashed line) are also shown. The frequencies used in the multiple sinusoidal fits are marked with vertical solid lines.


[TABLE]

Table 2. Amplitudes of the modulations for the UV continuum, optical and UV emission lines derived from a multicomponent sinusoidal fit.
Notes:
(1) Continuum flux amplitudes are in units of [FORMULA] ergs cm- 2s-1Å-1 while line fluxes are in units of [FORMULA] ergs cm-2s-1. Errors in parentheses are referred to the last significant digits.
(2) [FORMULA] is the average amplitude of the DFT of the residuals, calculated for frequencies [FORMULA]1.4 mHz.
(3) For the [FORMULA] 2900-2985 band only the orbital amplitude is reported as derived from IUE data.


A further check has been performed using the CLEAN algorithm (Roberts et al. 1987) which removes the windowing effects of the HST orbit. The CLEANED power spectra, adopting a gain of 0.1 and 500 iterations, indeed reveal the presence of the [FORMULA] and [FORMULA] sidebands in the continuum and the [FORMULA] in the emission lines. The lack of significant power at the other frequencies is consistent with the previous analysis. Hence, these weakly active frequencies will be considered with some caution in this analysis.

A strong colour effect in the UV continuum is detected with amplitudes decreasing at longer wavelengths. Different from other lines is the behaviour in the [FORMULA] feature, whose absorption component is modulated at the [FORMULA] frequency, while a variability at the spin is at a 2[FORMULA] level. No significant variations are detected in the equivalent width of the absorption as well as in the emission component.

In contrast to the flux behaviour, the V/R ratios are variable only at the [FORMULA] frequency (Fig. 2, bottom panels). Noteworthy is the marginal spin variability in the C IV line. The amplitudes of the spin modulation, obtained from the least-square fits, are reported in the last column of Table 2.

4.2. Optical data

The analysis of BVRI data acquired during three nights has been performed following the same procedure adopted for the HST data. Contrary to previous observations, the orbital variability also dominates in the optical being [FORMULA] 1.5 times the spin modulation. The presence of sideband modulations is more uncertain because of the lower quality of the data. Nevertheless, to ensure uniformity between UV and optical results, a least-square fit to the data has been applied using the same five frequency sinusoidal function, i.e. [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA]. The resulting amplitudes, reported in Table 2, when compared with the noise in the residuals (Column 8), can be considered as upper limits.

As far as the fast photometry is concerned, the low quality of the data only allows the detection of the spin and orbital variabilities. In particular, the latter is detected on the first night with a pronounced dip which is not consistent with the refined orbital ephemeris based on orbital minima recently given by Patterson et al. (1998), which defines the inferior conjunction of the secondary star. Therefore these data will not be used for a multi-wavelength analysis of the pulsations.

From this analysis new times of maxima for the orbital and rotational modulations are derived for the UV continuum and optical light:

  • [FORMULA] in the UV;

  • [FORMULA] in the UV;

  • [FORMULA] in the optical;

  • [FORMULA] in the optical;

Both UV and optical orbital maxima lead by [FORMULA] = 0.145 those predicted by Patterson et al.'s ephemeris. Such phase difference, discussed in Sect. 7, is consistent with the previous UV results (Paper 1).

On the other hand, the optical rotational maximum agrees within 8 per cent with that predicted by the new revised cubic ephemeris given by Patterson (1998, private communication):

[EQUATION]

The UV rotational pulses lag the optical by [FORMULA]. Such colour effect will be discussed in more detail in Sect. 5.

Furthermore, the time of coherence between spin and beat modulations is found to be [FORMULA] = 2 449 971.2335 [FORMULA] 0.0070 in both UV and optical. Throughout this paper the Patterson et al.'s orbital ephemeris will be used but [FORMULA] = 0.0 will refer to the orbital maximum. Hence, phase coherence occurs at [FORMULA]0.03. i.e. close to the orbital maximum, whilst in 1988 and 1990 it was found close to the orbital minimum (Osborne & Mukai 1989; Paper 1). Such phase changes are not uncommon for FO Aqr (Semeniuk & Kaluzny 1988; Hellier et al. 1990).

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© European Southern Observatory (ESO) 1999

Online publication: October 4, 1999
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