3 Data analysis and results

3.1 The average spectrum

The single typical spectrum and the average spectrum of FS Aur are shown in Fig. 1. The mean spectrum is an average of all spectra, corrected for wavelength shifts due to orbital motion. The spectrum is typical for a dwarf nova. It is dominated by strong and broad emission lines of hydrogen and neutral helium. In addition to them HeII $\lambda4686$ and weak CIII/NIII blend are observed also. All emission lines are single-peaked. The Balmer decrement is flat, indicating that the emission is optically thick as is normal in dwarf novae. There is no evidence of a contribution from a late-type secondary star. The equivalent width, FWHM, FWZI, and Relative Intensity of the major emission lines are presented in Table 1.

Figure 2: The variation of the H$\beta$ line profile around the orbit. The presence of a broad base component and a narrow line component is evident. There are orbital variations in the line profile. The narrow component remains practically symmetrical throughout an orbital cycle, but on the blue wing of the broad component at phases about 0.75-0.9 there appears a hump.

3.2 Emission lines variations

In order to increase the signal to noise ratio of the spectra we have phased the individual spectra with the orbital period, derived in the next section, and then co-added the spectra into 15 separate phase bins. Figure 2 shows the H$\beta$ profiles from some of the obtained spectra. We note a broad base component and a narrower line component which is presented throughout the orbital cycle. There are orbital variations in the line profiles. But whereas the narrow component remains practically symmetric throughout the orbital cycle, on the blue wing of the broad component at phases about 0.75-0.9 a hump appears. A similar behavior is observed for all lines, but most notably for H$\beta$.

For a more accurate examination of the profiles for asymmetry we calculated the degree of asymmetry of all profiles as the ratio between the areas of the blue and the red parts of the emission line. This parameter is very similar to the V/R ratio for double-peaked emission lines. It strongly depends on the wavelength range of the line wings which was selected for calculation of the degree of asymmetry. Choosing a greater wavelength range, we can analyze the farther line wings.

We calculated the degree of asymmetry for two values of the wavelength range of the line wings, and plotted this parameter as a function of orbital phase (left-hand frame of Fig. 3, upper panel: for a wavelength range of 13 Å, and lower panel: for a wavelength range of 30 Å). In the right-hand frame of Fig. 3 are shown some profiles of the H$\beta$line in which the wavelength range used for the determination of the degree of asymmetry has been marked. One can see that the central narrow line component indeed remains practically symmetrical throughout the orbital period, showing only a slight skewness to the right from time to time (Fig. 3, top). At the same time the broad base component of the line shows strong variability, and it becomes most asymmetric at a phase of about 0.9 (Fig. 3, bottom). This variability seems to contain information on the structure of the accretion disk and give evidence for an unusually located emission region.

Figure 3: The degree of asymmetry of the emission line H$\beta$ folded on the adopted period of 0 $.\!\!^{\rm d}$0595. The degree of asymmetry is the ratio between the areas of blue and red parts of the emission line. This parameter is very similar to the V/R ratio for double-peaked emission lines. The degree of asymmetry was calculated for two values of the wavelength range of the line wings (left-hand frame, upper panel for a wavelength range of 13 Å, and lower panel for a wavelength range of 30 Å). In the right-hand frame are shown some profiles of the H$\beta$ line in which the wavelength range used for the determination of the degree of asymmetry has been marked. One can see that the central narrow line component remains practically symmetrical throughout an orbital period, showing only slight skewness to the right from time to time. The broad base component of the line shows strong variability, and it becomes quite asymmetric at a phase of about 0.9.

3.3 Check of the period

Our photometric and spectral observations were carried out non-simultaneously and their short duration does not allow us to make any reliable period search. However as until now the orbital period of FS Aur was based only on spectra data (TPST), we have decided to check if our photometric data agree with spectral period.

Both our 2.1-hour and 4-hour light curves in the V-filter (Fig. 4) clearly show medium-amplitude (0 $.\!\!^{\rm m}$3) variations, although the flickering makes the light curves noisy. Although one can see a constant difference between two curves, their shape remains similar. However, no periodic modulations with spectroscopic period were detected, and thus the period of photometric modulations should be at least 3 hours.

As we detected a discrepancy between the photometric modulation period and the H$\alpha$ velocity variations period (from TPST), we decided also to check the spectral period using our spectral data. To verify the orbital period we measured the radial velocities in the H$\beta$ and H$\gamma$ emission lines using a Gaussian fit. To obtain an estimate of the period, a sine curve fit was made to the velocities, giving $P=0\hbox{$.\!\!^{\rm d}$ }059 \pm 0\hbox{$.\!\!^{\rm d}$ }002$ for H$\beta$ and H$\gamma$(Fig. 5). This period agrees well with the 0 $.\!\!^{\rm d}$0595 $\pm$ 0 $.\!\!^{\rm d}$0001 estimate by TPST. Additional evidence for this period, though less reliable, comes from the cyclic variations of the degree of asymmetry of the emission lines and their equivalent widths obtained from individual spectra, which oscillate with the same period.

Though we obtained different results based on photometric and spectral data, there are no reasons to doubt the reliability of the orbital (spectroscopic) period. Previous photometry of FS Aur over a 1.8-hour time span showed a 0 $.\!\!^{\rm m}$15 modulation in the B-filter with a period between 87-105 min (Howell & Szkody 1988) that is consistent with the spectroscopic period. Many CVs are known, for which the photometric behaviour varies for a short time. It is necessary to perform longer photometric observations of FS Aur during a single night to fully clarify this vagueness.

Note that henceforth we shall be using the period found by TPST because of its higher accuracy.

   
3.4 The radial velocity curve

In cataclysmic variables the most reliable parts of the emission line profile for deriving the radial velocity curve are the extreme wings. They are presumably formed in the inner parts of the accretion disk and therefore should represent the motion of the white dwarf with the highest reliability.

The velocities of the emission lines were measured using the double-Gaussian method described by Schneider & Young (1980) and later refined by Shafter (1983). This method consists of convolving each spectrum with a pair of Gaussians of width $\sigma$ whose centers have a separation of $\Delta$. The position at which the intensities through the two Gaussians become equal is a measure of the wavelength of the emission line. The measured velocities will depend on the choice of $\sigma$ and $\Delta$, and by varying $\Delta$ different parts of the lines can be sampled. The width of the Gaussians $\sigma$ is typically set by the resolution of the data.

Figure 4: Light curves of FS Aur on February 12, 1997 (closed circles, bottom X axis) and October 11, 1997 (open circles, top X axis).

Figure 5: The radial velocity curve of the H$\beta$ and H$\gamma$ emission lines, derived from a single Gaussian fitting method. To obtain an estimate of the period, a sine curve fit was made to the velocities, giving $P=0\hbox{$.\!\!^{\rm d}$ }059 \pm 0\hbox{$.\!\!^{\rm d}$ }002$ for both H$\beta$ and H$\gamma$.

Figure 6: The diagnostic diagram for the H$\beta$ (left panel) and H$\gamma$(right panel) data, showing the response of the fitted orbital elements to the choice of the double-Gaussian separation. The best fit is reached with Gaussian separations of $\approx$1500 kms-1 for H$\beta$ and of $\approx$1300 kms-1 for H$\gamma$.

We have measured the velocities in our binned spectra for the four emission lines (H$\beta ,$ H$\gamma$, HeI $\lambda$4471 and HeII $\lambda$4686) separately in order to test for consistency in the derived velocities. We measured the radial velocity using Gaussian separations ranging from 50 kms^-1 to 3200 kms^-1. All measurements were made using $\sigma =200$ kms^-1 and $\sigma =300$ kms^-1. For each value of $\Delta$ we made a non-linear least-squares fit of the derived velocities to sinusoids of the form

$$V(t,\Delta )=\gamma (\Delta )+K(\Delta )\sin \left[ 2\pi \left( T-T_0\left( \Delta \right) \right) /P+\pi\right]$$ (1)

where $\gamma$ is the systemic velocity, K is the semi-amplitude, T₀ is the time of inferior conjunction of the secondary star and P is the orbital period. The resulting "diagnostic diagram'' for H$\beta$ and H$\gamma$ with $\sigma =200$ kms^-1 is shown in Fig. 6. The diagram shows the variations of K, $\sigma(K)/K$ (the fractional error in K), $\gamma$ and T₀ with $\Delta$ (Shafter et al. 1986). The diagram for $\sigma =300$ kms^-1 looks the same.

To derive the orbital elements of the line wings we took the values that correspond to the largest separation just before $\sigma(K)/K$ shows a sharp increase (Shafter & Szkody 1984). Note that the dependence of parameter $\sigma(K)/K$ on Gaussian separation for both hydrogen lines is very similar (Fig. 6). For H$\beta$ it appears that $\Delta$ can be increased to $\sim$1500 kms^-1 before $\sigma(K)/K$ begins to increase. Similarly, the optimum value of $\Delta$ for H$\gamma$probably lies near 1300 kms^-1. Referring to the diagnostic diagram, the K values for H$\beta$, H$\gamma$, He I $\lambda$4471 and He II $\lambda$4686 are 73, 73, 81 and 69 kms^-1, respectively. The measured parameters of the best fitting radial velocity curves are summarized in Table 2. In Fig. 7 we show the radial velocity curves of H$\beta$ and H$\gamma$ emission lines.

 

 

Table 2: Elements of the radial velocity curves of FS Aur.

Emission	$\gamma$-velocity	K1	T0
line	(kms-1)	(kms-1)	(HJD)
H$\beta$	$46\pm 7$	$73\pm 10$	$2450101.264\pm0.001$
H$\gamma$	$68\pm 11$	$73\pm 15$	$2450101.265\pm 0.001$
HeI $\lambda$4471	$113\pm 10$	$81\pm 15$	$2450101.263\pm 0.002$
HeII $\lambda$4686	$50\pm 21$	$69\pm 29$	$2450101.265\pm 0.004$
Mean	$52\pm 7$	$73\pm8$	$2450101.264\pm0.001$

Basically, the radial velocity semi-amplitudes of the Balmer and Helium lines are consistent, while the $\gamma$-velocities are not. The reason for this is unknown. In the discussion to follow we will adopt a mean value of K and T₀ for these lines (using the $\sigma(K)$ and $\sigma(T_{0})$ as a weight factor): $K=73\pm8$ kms^-1 and $T_{0}({\rm HJD})=2450101.264\pm0.001$.

3.5 Equivalent width

We measured the equivalent widths of the H$\beta$ emission line in all individual spectra. They were investigated for a modulation with orbital period (Fig. 8). The errors have been obtained by calculating standard deviations from several independent measurements of the same lines.

Though the obtained equivalent widths exhibit rather significant dispersion, none the less one can see their modulation with orbital period. It is especially noticeable on the smoothed graph which was obtained by averaging adjacent data points (filled circles with dotted line in Fig. 8). We found that the EW is modulated with amplitudes not less than 25% of the mean value. We can confidently assert that there is a broad minimum in the EW around phase 0.1, and probably there is a secondary minimum near phase 0.6.

The observed minima could be due to an increase in the continuum luminosity when the enhanced emission region crosses the line-of-sight. If this is so then Fig. 8 testifies about complex and unusual accretion structure in FS Aur.

Figure 7: The H$\beta$ (closed circles) and H$\gamma$ (open circles) radial velocities measured using the double-Gaussian method folded on the orbital period. All data are plotted twice for continuity. The K-velocity is $73\pm 10$ kms-1 and $73\pm 15$ kms-1 respectively.

   
3.6 Possible system's parameters

It is impossible to measure the components' masses and the orbital inclination in a non-eclipsing, single-lined spectroscopic binary like FS Aur. However, we can determine preliminary values for the basic system's parameters, using the assumption that the secondary is a zero-age main-sequence (ZAMS) star (Patterson 1984) and that the secondary fills its Roche lobe. First of all, from our spectroscopic and photometric data we can limit the range of possible solutions.

Our observations reveal no evidence for eclipses, so we expected the inclination to be less than 65$^{\circ }$. Now, from the mass function for the system:

$$\frac{M_{2}\sin ^{3}i}{(1+q^{-1})^{2}}=\frac{PK^{3}}{2\pi G}$$ (2)

and any mass-period relations for the secondary stars we can obtain a lower limit to the mass ratio. Using any of the recently obtained empirical relations (see, for example, Caillault & Patterson 1990; Warner 1995; Smith & Dhillon 1998) we obtain a mass for the secondary less than $0.1~M_{\odot}$. Now, using Eq. (2) we find q>0.22 and, consequently, $M_{1}<0.46~M_{\odot}$.

Figure 8: The variation of the H$\beta$ equivalent widths with orbital period. Open circles show individual values, filled circles with dotted line show data which were obtained by averaging adjacent data points. The data are plotted twice for continuity.

Also we can place a stringent lower limit on M₁, assuming that the largest velocity in the emission line profile (FWZI/2) does not exceed the Keplerian velocity at the surface of the white dwarf:

$$\left( \frac{FWZI}{2\sin i}\right) ^{2}\frac{R_{1}}{G}<M_{1}$$ (3)

FWZI is not a well defined quantity because of the difficulty in establishing where the high velocity line wings end and the continuum begins. Besides, extensive wings due to Stark broadening can produce a spurious enhancement of emissivity at small radii. However, for the time being we assume that Stark broadening is not significant. The Balmer data show that the line wings extend to at least 1750 kms^-1from the line center. Using FWZI/2=1750 kms^-1, inclination angle $i<65^{\circ }$ and the mass-radius relation for white dwarfs (Hamada & Salpeter 1961),

$$\frac{{R_1 }}{{R_{\odot}}} = 0.0072\left( {\frac{{M_1 }}{{M_{\odot}}}} \right)^{ - 0.8},$$ (4)

we obtained $M_{1}>0.34~M_{\odot}$. And finally, using the last constraint on M₁ and Eq. (2), we can obtain a lower limit to the inclination: $i>51^{\circ }$. Table 3 summarizes all our calculated parameters.

 

 

Table 3: Adopted system's parameters for FS Aurigae.

Parameter	Value
T0 (Spectroscopic Phase 0.0)	$2450101.264\pm0.001$
Primary mass M1 ($M_{\odot}$)	0.34-0.46
Secondary mass M2 ($M_{\odot}$)	$\leq$0.1
Mass ratio q=M2/M1	$\geq$0.22
Inclination i	$51^{\circ}$- $65^{\circ}$