3 Results

3.1 Light curve. Orbital and spin modulations

The object shows multi-scale time variability with a range of 0.3 magnitudes (see Fig. 1). Four pronounced eclipse-like depressions obviously shape the light curve. Strong flickering with optical pulse amplitude (semi-amplitude) of about 0.1 magnitude is also obvious in the light curve detected and identified earlier (Israel et al. 1998; Uslenghi et al. 2000) as spin related modulations. The photometric data of 1WGAJ1958.2+3232 were analyzed for periodicities using the Discrete Fourier Transform (DFT) code (Deeming 1975) with a CLEAN procedure (Roberts et al. 1987). The CLEANed power spectrum (Fig. 2) of $R_{\rm c}$ photometric data shows a clear peak at $\Omega_{\rm orb}=5.54996\pm0.39624\ {\rm (1/day)}$, corresponding to $P_{\rm orb}=0.1802\pm0.0065{\rm d}$. This peak is caused by the above mentioned eclipses in the light curve and clearly marks the orbital period of the system.

We also found a significant peak at the spin period $\omega _{\rm rot}$ of the WD corresponding to $733.82\pm 1.25$ s. This period is in excellent agreement with that recently discovered by ASCA X-ray pulsations (Israel et al. 1999). The beat frequencies at $\omega _{\rm rot}-\Omega _{\rm orb}$, $\omega _{\rm rot}+\Omega _{\rm orb}$ are also present in the CLEANed power spectrum but with a smaller number of iterations (see insert in the upper right corner of the Fig. 2). The harmonics of the basic frequencies $\Omega _{\rm orb}$ and $\omega _{\rm rot}$ are detected as well. Besides these, there are comparably significant peaks at the periods of 727.78 s and 1.36 h. The former was detected also by Uslengi et al. (2000) and is probably the one-day alias of $\omega _{\rm rot}$, while for the latter we could not find any reasonable explanation.

Figure 3: The diagnostic diagrams for the ${\rm H_{\beta }}$ emission lines. The radial velocity semi-amplitude K, the ratio $\sigma _{\rm K}/K$ are plotted as a function of the Gaussian separation, obtained for a period of 0.18152 days

Figure 4: The radial velocity curves for the emission lines of H$_\beta$1 and He1II 4686 phased on the spectroscopic orbital period (0.18152 days) are presented upper and middle panels respectively. The filled circles corresponds to 4 Aug., filled rectangulars and open diamonds are 5 and 6 Aug., respectively. Binning time is 700 s. $R_{\rm c}$ light curve of 1WGAJ1958.2+3232 is presented in the lower panel. The binning time is 120 s

Figure 5: The power spectrum of the $R_{\rm c}$ light curve and the He1II 4686 and H$_\beta$1 radial velocity curves are presented. They are scaled to the amplitude of the $R_{\rm c}$ power spectrum. The maximum peak of frequency corresponds to the orbital period of the system $P_{\rm orb}=0.18152\pm 0.0011{\rm d}$

3.2 Radial velocity variations and binary system parameters

The spectrum of 1WGAJ1958.2+3232 shows features characteristic for Cataclysmic Variables. We refer to Negueruela et al. (2000), who obtained a spectrum of the object in a wider spectral range and with better spectral resolution. We also obtained time-resolved spectroscopy of 1WGAJ1958.2+3232 around the emission lines of H$_\beta$1#912#> and He1II, covering several orbits. Thus, we were able to examine periodical variations in the spectrum of the object, primarily in the emission lines. The simple stacking of consecutive spectra onto the trailed spectrum showed strong variability in the lines. It is distinct in the Balmer lines and in the higher excitation lines of ionized Helium. The Balmer lines are double-peaked with an S-wave moving inside, which makes it hard to see the periodic pattern. In the He1II 4686 line the central narrow component dominates in most of the phases, and it shows clear sinusoidal variation.

In order to determine the orbital elements we measured the radial velocities (RV) of H$_\beta$1 applying the double Gaussian deconvolution method introduced by Schneider & Young (1980), and further developed by Shafter (1983). This method is especially efficient for measurements of the orbital motion of CVs with a prominent spot at the edge of the accretion disk, contaminating the central parts of the emission lines. It allows us to measure RV variations using the wings of the lines. The width of the Gaussians were chosen to be slightly larger than our spectral resolution (8.5 Å), where deconvolution was reached at all orbital phases. The radial velocities were measured as a function of distance a between the Gaussians, and then the diagnostic diagrams were constructed using an initial guess for the orbital period, derived from photometry and from preliminary radial velocity measurements via Gaussian fits to the lines. The optimal value of separation (a= 1175 km ${\rm s}^{-1}$) was determined from the diagnostic diagrams, and the RV values measured for these Gaussian separations were again subjected to a power spectrum analysis in order to refine the period. The spectroscopic period peaked at a slightly longer value, than the photometric period (however within the errors of the photometric period). This method quickly converged and after two iterations no further improvement was achieved. The diagnostic diagrams for H$_\beta$1 are shown in Fig. 3. Figure 4 (top) shows the H$_\beta$1 radial velocity curve.

Figure 6: The inclination angle of the system vs. q=M2/M1 is shown (solid line). The mass of WD in the system M1 vs. q for M2=0.41 $M_{\odot }$ is given by the dot-dashed line. "Best fit'' parameters from the secondary star position and the ballistic trajectory of the gas stream on the Doppler maps are marked

A narrow single Gaussian profile was fitted to the prominent emission features in the profile of He1II $\lambda 4686$ Å, and the measured line centers were used to determine the radial velocity solution for 1WGAJ1958.2+3232. The radial velocity curve for He1II $\lambda 4686$ line is presented in the middle panel of Fig. 4. These measurements were also subjected to the power spectrum analysis. The obtained orbital period is in good agreement with the values derived from the photometry and the H$_\beta$1 line radial velocities. The power spectra around the values corresponding to the orbital period from these three independent determinations are plotted in Fig. 5. One can see the excellent match of the central peak. We adopted $0\hbox{$.\!\!^{\rm d}$ }18152\pm 0\hbox{$.\!\!^{\rm d}$ }00011$ as the final value for the orbital period of 1WGAJ1958.2+3232 from our observations. Longer time base observations are needed to improve this value.

Each of the radial velocity curves was fitted using a least-squares routine of the form

$$v(t) = \gamma_{{\rm o}}+K_{1}*\sin(2\pi(t-t_{\rm o})/P),$$ (1)

where $\gamma_{\rm o}$ is the systematic velocity of the system, and K₁is the semi-amplitude of radial velocity, both in km ${\rm s}^{-1}$. The observation time is t, the epoch $t_0=2451761\hbox{$.\!\!^{\rm d}$ }2281\pm0\hbox{$.\!\!^{\rm d}$ }0001$corresponds to the $\pm$ zero crossing of the ${\rm H_{\beta }}$ radial velocity curve, and therefore is a superior conjunction of the binary system (secondary located between observer and the WD). Accordingly the phase value at t₀ was set to 0.0. Table 2 gives a summary of the radial velocity fits for H$_\beta$1 and He1II 4686 emission line.

 

 

Table 2: Radial velocity solution parameters

Name	$\gamma_{\rm o}$	K1
	kms-1	kms-1
${\rm H_{\beta }}$	$-39\pm5$	$74\pm7$
${\rm He II 4686}$	$-70.2\pm 15.0$	$197.4\pm 25.7$

After refining the orbital period from spectroscopy, and determining the phase 0.0, the photometric light curve was folded by the corresponding parameters and presented in the lower panel of Fig. 4.

Figure 7: Trailed, continuum-subtracted, spectra of 1WGAJ1958.2+3232 plotted in two cycles. Doppler maps of the emission lines H$_\beta$1 (left panel), He1II (right panel) in velocity space (Vx,Vy) are given. A schematic overlay marks the Roche lobe of the secondary, the ballistic trajectory and the magnetically funneled horizontal part of the accretion stream. The secondary star and gas-stream trajectory are plotted for K=74km  ${\rm s}^{-1}$ and q=0.46

Several conclusions can be made after considering the three curves in Fig. 4 in conjunction with and taking into account common knowledge of Intermediate Polar systems (Patterson 1994; Warner 1995):

• The double peaked Balmer lines are due to the presence of an accretion disk or ring orbiting the primary WD in this IP. RV variations in the wings of lines describe the rotation of the primary of the binary system;
• The S-wave present in the Balmer and He1II emission lines is evidence for a hot compact region on the accretion disk;
• From the difference of phases and amplitudes of radial velocities of the wings of H$_\beta$1 and the narrow component of He1II we can conclude that the S-wave producing hot spot is located at $\approx$0.1-0.2 phase ahead of the superior conjunction. It is difficult to estimate the location more precisely, because the matter in the spot has an intrinsic velocity. It is the usual location of the spot originating from the impact of the mass transfer stream with the accretion disk. However it is also the area which is most commonly heated by the energetic X-ray beam from the magnetically accreting pole on the surface of the WD in IPs (see the sketch in Hellier et al. 1989);
• The hot spot itself is eclipsed by the accretion ring as follows from the phasing of the light curve, the dips in the light curve centered at the phase $\approx$0.12 $\phi_{\rm orb}$. The primary at this phase starts to move toward the observer and the hot spot is on the opposite side of the ring approaching maximum velocity.

From our spectroscopic radial velocity solution, we can determine preliminary values for the basic system parameters of 1WGAJ1958.2+3232. First, from the mass-period and radius-period relations of Echevarría (1983)

$$M_2/M_{\odot}$$ = 0.0751 P(h)^1.16,

$$R_2/R_{\odot} 0.101P(h)^{1.05}, \ 1.4< P(h)<12$$ = (2)

we estimate the mass and radius of the secondary star as M₂ = 0.41 $M_{\odot }$ and R₂=0.47 $R_{\odot}$.

On the other hand, we can constrain the relation between inclination angle i versus mass ratio q:

$$\sin^3(i)=\frac{K_1^3 P}{2\pi G M_2}\left(\frac{q+1}{q}\right)^2$$ (3)

if the mass ratio of the system is known (see Downes et al.  1986; Dobrzycka & Howel 1992). The dependence of i versus $q = M_2/M_{\rm WD}$ in the range 0.4 up to 0.75 is shown in Fig. 6 for the above determinated values of ${K_1}_{\rm H_{\beta}}$, $P_{\rm orb}$ and $M_{{\rm 2}}$.

Meanwhile, the mean mass estimate of 76 white dwarfs in CVs is $M_{\rm WD} = 0.86$ $M_{\odot }$ (Sion 1999). Webbink (1990) gives statistically average white dwarf masses ratios (q = 0.29) and average masses for all systems ( $M_{\rm WD} = 0.61~$$M_{\odot }$) below the period gap and (q = 0.64, $M_{\rm WD} = 0.82$ $M_{\odot }$) above the period gap. Thus, the possible solutions lie in the narrow range of values.

We attempted to refine these values for 1WGAJ1958.2+3232 by constraining Doppler tomograms from observed emission line profiles.

   

Table 3: The 1WGA1958.3+3232 adopted system parameters

Parameter	Value	Parameter	Value
$P_{\rm orb}$	0.18152d	R2	0.47 $R_{\odot}$
$P_{\rm rot}$	733.7s	a	1.5 $R_{\odot}$
M2	0.41 $~M_{\odot}$	i	$35^{\rm o}$
q	0.46	$M_{\rm WD}$	0.9 $~M_{\odot}$

Figure 8: Doppler maps of the emission line blend C1III/N1III in velocity space (Vx,Vy) is given. A schematic overlay marks the Roche lobe of the secondary, the ballistic trajectory and the magnetically funneled horizontal part of the accretion stream. The secondary star and gas-stream trajectory are plotted for K=74 km s-1and q=0.46

Doppler tomography is a useful tool to extract further information on CVs from trailed spectra. This method, which was developed by Marsh & Horne (1988), uses the velocity profiles of emission-lines at each phase to create a two-dimensional intensity image in velocity-space coordinates (V_x,V_y). Therefore, the Doppler tomogram can be interpreted as a projection of emitting regions in cataclysmic variables onto the plane perpendicular to the observers view. We used the code developed by Spruit (1998) to constrain Doppler maps of 1WGAJ1958.2+3232 with the maximum entropy method. The resulting Doppler maps (or tomogram) of emission lines of H$_\beta$1, He1II and the blend of C1III/N1III are displayed as a gray-scale image in Figs. 7 and 8. Also in Fig. 7 are displayed trailed spectra of H$_\beta$1 and He1II in phase space and their corresponding reconstructed counterparts. Two features in the maps are distinct: an accretion disk seen as a dark circle extending to up to $\sim$-700 kms^-1 on H$_\beta$ doppler tomogram and a bright spot detected in all three maps to the left and below the center of mass at velocities $V_x \sim -225~{\rm km\,s^{-1}}$, $V_y \sim -100~{\rm km\,s^{-1}}$in He1II 4686. Apart from the spot a cometary tail linked to it and extending to $V_x \approx V_y \approx -500~{\rm km\,s^{-1}}$ can be clearly seen on the He1II map. These we identify with the mass transfer stream and its shape was essential for our selection of the ballistic trajectory. A help in interpreting Doppler maps are additional inserted plots which mark the position of the secondary star and the ballistic trajectory of the gas stream. Here we used our estimates of $P_{\rm orb}$ and $M_{{\rm 2}}$ with various combinations of iand q from Fig. 6 in order to obtain the "best fit'' (by simple eye inspection) of the calculated stream trajectory with the gray scale image. Our preferred solution for the inclination is $i=35^\circ$. It is marked in Fig. 6 and given in Table 3. Of course other close solutions are applicable. Comparing the location of spots in Doppler maps of He1II and H$_\beta$1 we can actually distinguish two hot spots on the disk (Fig. 7). The elongated spot coinciding in both emission lines is probably caused by the mass transfer stream and the shock of impact with the disk, while the compact dense spot toward negative V_y's, seen much better in H$_\beta$1 than in the two other lines, is a result of heating of the disk by the X-ray beam. The C1III/N1III pattern mostly repeats that of He1II, with lower intensity though (Fig. 8).