A new radial velocity study of the well-known dwarf nova SS Cygni: a low-mass binary system

ABSTRACT

We present new high-dispersion spectroscopic observations of the cataclysmic variable SS Cygni at the end of an outburst (decline) and during the first days at quiescence. We determined the orbital parameters of the white dwarf (WD) and the red dwarf (RD) companion at quiescence. The mass functions at this stage are |$M_{wd} \sin ^3{i} = 0.24 \pm 0.01$| M|$_{\odot }$| and |$M_{rd} \sin ^3{i} = 0.11 \pm 0.01$| M|$_{\odot }$|⁠, and the separation is |$a \sin i = 1.25 \pm 0.01$| R|$_{\odot }$|⁠. Since this binary is not an eclipsing system, a direct value of the inclination i cannot be determined. However, published photometric curves and their model fitting yield the range |$45^\circ \le i \le 56^\circ$|⁠. Using this interval we obtain masses of the WD and RD within the ranges |$0.42\,\,\mathrm{M}_\circ \le M_{wd} \le 0.67$| M|$_{\odot}$| and |$0.19 \le M_{rd} \le 0.30$| M|$_{\odot }$|⁠, respectively, and separation in the range |$1.51 \le a \le 1.77$| R|$_{\odot }$|⁠. This is an important result, as our range of masses is lower than those previously published. We argue that our determination of the semi-amplitude |$K_{rd} = 158.6 \pm 0.7$| km s|$^{-1}$| is a solid one. However, this is not the case of |$K_{wd}$|⁠. The large scatter found in this parameter and the uncertainty in the inclination angle explain the large range in the masses published in previous works. Finally, we present Doppler tomography of several Balmer lines. We find that during the decline, the accretion disc is barely visible, showing mainly a hot spot, while during the minimum state a disc with a spiral-arm-like structure is observed.

1 INTRODUCTION

SS Cyg is one of the best observed dwarf novae (DNe) and indeed one of the brightest cataclysmic variables (CVs), which are interacting binaries consisting of a white dwarf (WD) primary star and a late-type main-sequence star (secondary), normally a red dwarf (RD) that transfers matter to the primary via Roche Lobe overflow (see Warner 1995, and references therein). The so-called classical model corresponds to a non-magnetic system with normal and frequent outbursts. Smak (1971) and Warner (1995) proposed that the material transferred from the secondary flows through the inner Lagrangian point and orbits around the primary forming an accretion disc. In particularly, DNe are systems without a strong magnetic field in the primary star, which enables an accretion disc to be fully formed. These systems, as mentioned above, have frequent outbursts from 2 to 5 mag and have an outburst frequency from |$\sim 10$| d to tens of years with a well-defined time-scale for each object (for a more detailed morphology of these systems see also Warner 1995). A historical review of the old observations of SS Cyg can be found in Gaposchkin (1957) and Zuckermann (1962). Earlier values of the orbital period 0.276 d and a spectral type of dG5 were first determined by Joy (1956). The best determinations of the orbital period and spectral type are 0.27512973 d (Hessman et al. 1984) and K5V (North et al. 2002).

The first radial velocity study was done by Joy (1956) with photographic plates. There have been many radial velocity studies since this pioneering work. These are summarized in Section 5 and Table 4. As we will show in Section 5 there is a good agreement between the various works about the value of the semi-amplitude of the RD. This is not the case for the |$K_{wd}$| value, which shows a large range from about 80 to 120 km s|$^{-1}$|⁠. Due to the variety of methods used to determine this value (see Section 5), we decided to embark on a new and modern radial velocity study of this well-known and bright DN.

Although a double-lined binary, it is not an eclipsing system and therefore its masses cannot be determined directly. The WD is not directly visible in the optic, and only the emission lines from the accretion disc are visible. Therefore, assumptions are usually made that the semi-amplitude of the radial velocity curves of the emission lines, |$K_1$|⁠, represent the motion of the primary star, i.e. |$K_1 = K_{wd}$|⁠, while |$K_2 = K_{rd}$| represents the motion of the secondary star.

Given that CVs are spectroscopic binaries, the study of the radial velocities of their components is paramount to obtain the orbital parameters, and therefore the masses of the system. These masses are important to study the evolution of the CVs.

Additionally, Doppler tomography, a method developed by Marsh & Horne (1988) to analyse the profile of the emission lines, was used to create two-dimensional velocity-space maps of the accretion disc.

2 OBSERVATIONS

In this work, spectra were acquired with the Echelle spectrograph¹ in 2015 September, with a resolving power of R |$\sim$| 15 000, at the f/7.5 Cassegrain focus of the 2.1 m telescope of the Observatorio Astronómico Nacional at San Pedro Mártir (OAN-SPM), b.c., México.

In total, 105 SS Cyg spectra of 900 s exposure each, were obtained over six nights (see Table 1), using the CCD Marconi 2 2048 |$\times$| 2048 pixels, which have a size of 13.5 microns. These spectra were reduced and treated with the image reduction and analysis facility software (iraf²). Spectra of the standard late-type stars 61 Cyg A and 61 Cyg B were also obtained.

We observed the object at the end of an outburst and part of the following minimum. This is shown in the SS Cyg light curve from the aavso for the month of 2015 September (see Fig. 1). Our observations fall into two different stages of the light curve. The first three nights were observed when the binary was in decline from an outburst, while the last three nights were taken at an early quiescent state. We will therefore divide our work into analysing the data in decline and early quiescence.

3 ANALYSIS AND RESULTS

3.1 Construction of complete one-dimensional spectra

We have constructed complete one-dimensional spectra of all the observed orders with the purpose, mainly, to take advantage of cross-correlating a larger number of absorption lines. In order to construct these one-dimensional spectra, we first proceeded to normalize all the available orders (orders 33 to 58) of both SS Cyg and our spectral standards. The object spectra will therefore contain both the absorption and Balmer emission lines. To do this, we first used the splot function of iraf, using the task normalize to adjust the continuum with polynomial functions of orders between 6 and 9 without taking into account the spectral lines. This allowed us to flatten the orders one by one. A similar procedure was used for our spectral standards 61 Cyg A and B. Once each order was normalized, they were combined to create a one-dimensional spectrum. The signal at the edges in each order is very poor due to the fact that the normalization function usually generates a lot of noise at the edges and produces unwarranted errors in the combination process. To avoid this problem, we cut these low signal regions before combining the orders with the scombine task, therefore creating a single spectrum from 3815 to 6860 Å. Fig. 2 exemplifies the final result for one of the SS Cyg spectrum obtained during the last night at quiescence.

3.2 Radial velocity analysis of the secondary star

To obtain the radial velocities of the secondary star, a cross-correlation method was used within the spectral range 3900–6800 Å. This method was applied using the fxcor task within the iraf rv package. The emission lines at SS Cyg were properly masked. A template spectrum with known and reliable heliocentric radial velocities was used. We selected 61 Cyg A, which is a well-studied K5V-type star defined as a standard in the Morgan–Keenan classification, whose velocity is |$-65.82 \pm 0.06$| km s|$^{-1}$| (Halbwachs, Mayor & Udry 2018). We have found that the fxcor task has an error in the HJD date output; it adds 1 d to the resulted calculations (a possible code error in the task). This has been corrected using the HJD values from the headers of the spectra.

Once the radial velocities were found, they were fitted with a circular orbit of the form

where |$\gamma$| is the systemic velocity, K the semi-amplitude, |$t_0$| the time of inferior conjunction of the donor, and |$P_{orb}$| is the orbital period. We employed |$\chi ^2$| as our goodness-of-fit parameter. The errors of the fitted variables are simply the statistical uncertainty in the least-square method used. It is pertinent to point here that the |$\chi ^2$| found for the 63 spectra used in the fit of the secondary has an equivalent |$\chi ^2_{\nu } = \chi ^2/\nu = 0.905$|⁠, where |$\nu$| equals 60 deg of freedom, which yields a standard deviation of |$\sigma =0.95$|⁠. Following the methodology carried out by Horne, Wade & Szkody (1986) the orbital period was fixed in equation (1), and therefore only the other three parameters were calculated. Our fits have been obtained by running orbital,³ a simple least square program, to determine the three orbital parameters with the fixed orbital period. We used here the well-established value of 0.27512973(2) d (Hessman et al. 1984). The results of the fit presented in Fig. 3 are from the last three nights using the 63 spectra observed at quiescence (see Table 1). Table 2 (second column) shows the orbital parameters calculated with the previous fit. The |$K_2$| calculated in quiescence is |$158.6 \pm 0.7$| km s|$^{-1}$|and |$\gamma = -12.1 \pm 0.5$| km s|$^{-1}$|⁠. The residuals at the bottom of this figure show that the data can be fitted, to a first order with a circular orbit. However, there are negative deviations around phase 0.25 and 0.60 and positive deviations around phase 0.45 and 0.90. These deviations are not sinusoidal. We have made a similar analysis for the nights during decline. The results are shown in Fig. 4. The data in this case are clearly not well fitted with a circular orbit, as we now observe larger deviations in the residuals at the same phases as before. These facts could well reveal that the secondary star is highly spotted, with strongly asymmetric irradiation on the inner hemisphere, and, as suggested by Hill et al. (2017) and Hessman et al. (1984), this will increase the semi-amplitude. In fact the |$K_2$| calculated in decline is |$180.0 \pm 3.3$| km s|$^{-1}$|⁠, which is greater that the |$K_2$| value obtained in quiescence. The |$\gamma$| velocity found at this stage is |$\gamma = -6.3 \pm 2.3$| km s|$^{-1}$|⁠.

3.3 Radial velocity analysis of the primary star

To measure the radial velocities of the emission lines and ascribe their orbital velocities to that of the primary star, i.e, take the important assumption that |$K_{WD}$| = |$K_1$|⁠, a method was proposed by Schneider & Young (1980) on the basic idea that the wings of the emission lines will be less prone to present asymmetries during the orbital phases and therefore, by measuring them, will satisfy the assumption |$K_{WD}$| = |$K_1$|⁠. To measure the wings of the emission lines, we used the iraf task convrv, which is embedded into the rvsao package (this task was given to us privately by J. Thorstensen in 2008). The task computes the velocity of a line for a set of spectra by convolving the line with an antisymmetric function and takes the line centre to be the zero of this convolution, based on the algorithms described by Schneider & Young (1980) and Shafter, Szkody & Thorstensen (1986). More details about convrv are given in Segura Montero, Ramírez & Echevarría (2020). This method, which uses two Gaussian functions with a fixed width a and variable separation s, has a final goal of producing a diagnostic diagram. In this diagram, we will look primarily for a minimum value of the control parameter, |$\sigma _{K}/K$|⁠, which is a very good indicator of the best K value (Shafter et al. 1986). Constructing a diagnostic diagram requires an interactive fitting between the convrv routine and the program orbital, described in Section 3.2, to determine the best orbital parameters. In our calculations, the orbital period remains fixed, as in the previous section.

Our values were obtained from an analysis of the Balmer lines during the nights at quiescence, since during decline the accretion disc is barely visible as it is in the upper part of the surface density versus disc temperature thermal instability diagram (e.g. Hellier 2001, section 5.3). Prior to this, we have subtracted the secondary star using broadened versions of 61 Cyg A corrected for the radial velocity of each SS Cyg spectrum. A full explanation of this procedure can by found in Echevarría et al. (1989). The results for H|$\alpha$|⁠, derived from Figs 5 and 6, show a good fit; however, the |$\gamma$| velocity (⁠|$-49.9 \pm 1.0$|⁠) is much higher in module than that found for the secondary star. The equivalent results for H|$\beta$|⁠, shown in Figs 7 and 8, and H|$\delta$|⁠, shown in Figure 9, give noisier values. The |$K_1$| result is consistent, within the errors, with those of H|$\alpha$|⁠; see Table 2. Here, the |$\gamma$| velocity (⁠|$-29.2 \pm 1.2$|⁠) is closer to that found for the secondary star, but is still too higher in module. These discrepancies are addressed in the Discussion. The |$H_{\gamma }$| line was also subtracted using the method described above. However, a |$\sigma _{K}/K$| minimum could not be found. The results, shown in the last column of Table 2, have been derived using the separation and width for H|$\beta$|⁠. It is clear that these results lack our formal criteria of obtaining a |$\sigma _{K}/K$| minimum and the orbital parameters solution is shown mainly to illustrate how the noise from this line becomes greater than the H|$\alpha$| and H|$\beta$| ones. Using this line will be useful in the Doppler tomography analysis (see Section 4).

3.4 Orbital parameters

The velocity of a binary component and the orbital period provide information about the separation and gravitational force between the two components and therefore about the masses. To obtain the masses, we base ourselves on the results obtained from our calculations of the radial velocity of both components. The resulting mass functions using the values obtained for |$K_2$| and the mean value of |$K_1$|⁠, for H|$\alpha$| and H|$\beta$| (⁠|$K_1 = 71.4 \pm 1.6$|⁠) in quiescence state, are

while the separation between the two stars is given by

With respect to the angle of inclination, we adopt the range of values |$i = 45^{\circ }$| to |$i = 56^{\circ }$| obtained by Bitner et al. (2007) adjusting light curves of the object to the minimum obtained by Voloshina & Khruzina (2007). This range is consistent with the lower value |$i = 45^{\circ }$| adopted by Hill et al. (2017), obtained from simultaneous photometry to their spectroscopy (see their fig. 3.) Table 3 shows the masses and separation calculated as a function of this range for the inclination angles. Using the extreme values of i and our obtained |$K_1$| and |$K_2$| values, for both the secondary component and the mean value of H|$\alpha$| and H|$\beta$| at quiescence, we obtain masses |$0.42\,\,\mathrm{M}_\circ \le M_1 \le 0.67$| M|$_{\odot }$|⁠, |$0.19\,\,\mathrm{M}_\circ\le M_2 \le 0.30$| M|$_{\odot }$|⁠, and a separation |$1.51\,\,\mathrm{R}_\circ \le a \le 1.77$| R|$_{\odot }$|⁠.

4 DOPPLER TOMOGRAPHY

Doppler tomography is a spectroscopy technique that uses the phase-resolved emission-line profiles to map the accretion flow in velocity space, using the phase-resolved profiles of the spectral emission lines. A detailed formulation of this technique can be found in Marsh & Horne (1988).

In this work we used the lines of |$H\alpha$|⁠, H|$\beta$|⁠, and H|$\gamma$|⁠. The Doppler tomographies were obtained for the decline and quiescence nights separately since we want to study the nature of the accretion disc in these two stages. To obtain the tomography, the program pydoppler  ⁴ developed by Hernandez Santisteban (2021) was used. For each case, the trailed spectra were obtained and with it the Doppler tomography was constructed. In addition to this, a reconstruction of the trailed spectra is made from the obtained Doppler tomography. In total, six Doppler tomographies were obtained, three for the decline nights and three for the quiescence, shown in Figs 10 and 11, respectively.

The three tomograms from the decline nights show all a similar behaviour. We see a hot spot close to the inner Lagrangian point |$L_1$|⁠. We observed that this spot moves slightly to higher |$V_x$| velocities as we look from H|$\alpha$| towards H|$\gamma$|⁠. The accretion disc is barely visible as it is in the upper part of the surface density versus disc temperature thermal instability diagram, but we have highlighted it by increasing the low-cut of the stream program.

From the three quiescence nights, which were obtained right at the end of an eruption, we observe what looks like a spiral arm-type structure. This is consistent with previously reported tomography of the object (Kononov et al. (2012; Hill et al. 2017), whereby an asymmetric structure overlapping the disc is clearly visible. Our tomography in the decline phase also shows a suggestive asymmetric structure, but it is possible that the overall high temperature of the disc hides the asymmetry. The regions of maximum emission are found at low velocities in H|$\alpha$|⁠, H|$\beta$|⁠, and H|$\gamma$| corresponding to the hot spot. In addition, an asymmetric emission (the red zone) is observed in H|$\alpha$| at |$-100\,\,\mathrm{ km}\,\mathrm{ s}^{-1} \lesssim V_x \lesssim 200$| km s|$^{-1}$| and |$50\,\,\mathrm{ km}\,\mathrm{ s}^{-1} \lesssim V_y \lesssim 300$| km s|$^{-1}$|⁠. In H|$\beta$| there is an arm-shaped region that extends at positive |$V_x$| velocities in a range of |$-50\,\,\mathrm{ km}\,\mathrm{ s}^{-1} \lesssim V_x \lesssim 350$| km s|$^{-1}$| and |$-100\,\,\mathrm{ km}\,\mathrm{ s}^{-1} \lesssim V_y \lesssim 300$| km s|$^{-1}$|⁠. In H|$\gamma$| there is an extensive red zone with similar behaviour to that in H|$\beta$| whose arm extends in negative |$V_x$| in the range of |$-400\,\,\mathrm{ km}\,\mathrm{ s}^{-1} \lesssim V_x \lesssim 450$| km s|$^{-1}$| and |$-500\,\,\mathrm{ km}\,\mathrm{ s}^{-1} \lesssim V_y \lesssim -400$| km s|$^{-1}$|⁠. In standard CV discs, the temperature increases inwards. Therefore, spectral lines with higher excitation energies should originate in regions closer to the accretor. For Keplerian discs, the velocity also increases inwards, therefore lines with higher excitation energies (up to H|$\gamma$|⁠) in our tomograms show us regions at higher velocities, meaning that for the higher transitions we are observing deeper layers of the disc, similar to the results of Kononov et al. (2012).

5 DISCUSSION

We note that the error associated with the semi-amplitude |$K_2$| is small, as are the residuals shown in Fig. 3. We think that it is the result of using the one-dimensional spectra as in our constructed spectra we used a very large number of absorption lines to perform the cross correlation, masking the Balmer and He i emission lines. Therefore, we will use this value of |$K_2$| as the cornerstone of the results in this paper. As we will note in the following discussion, there is a general agreement with other published results. Robinson et al. (1986) have noted that among the published semi-amplitudes, |$K_2$| fall into two groups, one around 120 km s|$^{-1}$| and the other one near 155 km s|$^{-1}$|⁠. Since the first group depends mainly on the observations by Joy (1956), Robinson et al. decided to revise the results of Cowley et al. (1980) and Walker (1981). In the case of Walker results, they remeasured their plates again. The semi-amplitude |$K_2 = 146 \pm 7$| km s|$^{-1}$|⁠, which they found, is much higher than the velocity reported by the author. They attribute the discordant values to the use of different spectral lines to measure the semi-amplitude. They argue that the lines used by Walker do not constitute a good set of lines for CVs (e.g. contamination from other lines, weak or saturated lines, etc.). Concerning the differences in |$K_2$| obtained by Cowley et al. they could not attribute their lower velocity to differences in the measurement method or to the choice of spectral lines because they used a similar measuring machine and used the same spectral lines. Robinson et al. points out, from a speculation by Cowley et al., that a possible source of the differences between the various |$K_2$| measurements is the heating of the secondary star by the WD and its accretion disc. To test this hypothesis, Robinson et al. performed a more quantitative estimate of the effects of heating on the radial velocity curve by creating synthetic radial velocity curves using a light-curve synthesis program adopting the standard model for non-magnetic CVs. After making a small correction for the effects of residual heating and geometric distortions, they find that the |$K_2$| velocity of the centre of mass of the secondary star is |$158 \pm 3$| km s|$^{-1}$|⁠. After the paper Robinson et al. there have been several publications where |$K_2$|⁠, though calculated with different methods, hovers around this value of 158 km s|$^{-1}$|⁠, as shown in Table 4.

Since the radial velocity semi-amplitude of the secondary star is now well established, it is evident that a good determination of |$K_1$| is essential to determine the masses of the binary. Since this system is non-eclipsing, a reasonable estimation of the inclination angle is also necessary. If we compare in Table 4 the semi-amplitudes of the primary component obtained from different authors, we see that the measurements have a large scatter, ranging from 71 to 118 km s|$^{-1}$|⁠. We discuss now the different determinations of |$K_1$| and the inclination angle assumed in each case. In the pioneering work of Joy (1956) he obtains a value of 115 km s|$^{-1}$| (no errors given). His analysis is based on photographic plates and an orbital period is established for the first time. The 31 measurements were done at minimum light during five different years spanning 1941 to 1955. Joy specifies that the velocity measures of both dark and bright lines are exceedingly difficult and are subject to large errors. The velocities shown in his diagram represent three of four observations. The solid lines do not represent a formal solution. He argues that the lack of eclipses should imply |$i \le 60^\circ$|⁠. He takes the statistical value of |$50^\circ$| to derive the masses shown in Table 4, which are rather low, compared with most estimates from other works. Great lengths have been taken since, to explain the results of Joy; although his approach was not very formal, it constituted the first basic attempt to measure |$K_1$|⁠, |$K_2$|⁠, and to calculate the masses of the binary. The first work where the wings of H|$\alpha$| were measured was made by Kiplinger (1979), using a Grant Comparator, and only a part of the wings was selected. This technique tried to minimize the asymmetrical part of the emission lines that should be present at low velocities, i.e. the line centre. As we saw in Section 3.3 this idea has been developed further using the algorithms described by Schneider & Young (1980). Kiplinger adopted a very low value of i = |$30^\circ$|⁠, resulting in high mass values. The measurements by Stover et al. (1980) were carried out with a Digicon Detector described in full by Tull, Vogt & Kelton (1979). They measured the wings of H|$\beta$|⁠, H|$\gamma$|⁠, and H|$\delta$| by fitting symmetric polynomials to the wings of the lines. Their result for |$K_1$| have a small error and their radial velocity curve has some troublesome points around phase zero. However, their estimation of the inclination angle, following a similar analysis as Kiplinger, lead to a very small inclination angle of |$i = 35^\circ$|⁠, which resulted in a mass for the primary close to the Chandrasekhar limit. However, their |$K_1$| estimate agreed, within the errors, with four other publications (see Table 4). An important work on the radial velocities of SS Cyg was made by Hessman et al. (1984). The object was observed both at minimum and during outburst. Due to the discrepancies in the values reported on |$K_1$| and |$K_2$|⁠, at the time of their publication, they sought the source of the discrepancies. We will not go further here, since seven more works have been published, and it is important to include them. Of particular importance in their paper is the conclusion that |$K_1 = K_{WD}$|⁠. Hessmanfind that the Balmer absorption lines during the eruption agree with those of the broad Balmer emission lines observed at minimum light, indicating that the emission lines are reasonably accurate indicate of the true orbital motion of the WD. Finally, based on several arguments, they find a consistent set of |$M_{WD} = 1.1$| M|$_{\odot }$|⁠, |$M_{rd}= 0.65$| M|$_{\odot }$|⁠, and |$i = 38^\circ$|⁠. As we will see at the end of this discussion, Hessman et al.’s assumption that |$K_1 = K_{WD}$| may not always be valid. Echevarría et al. (1989) using an Echelle spectrograph and a Mepsicron-type detector observed the system at the beginning of a minimum and measured the wings of H|$\beta$| and H|$\gamma$|⁠, subtracting the absorption spectrum of the secondary star and masking the central region of the lines and leaving only the high-velocity zones. From their analysis, they found |$K_1 = 96 \pm 5$| km s|$^{-1}$| and |$K_2 = 152 \pm 2$| km s|$^{-1}$|⁠. Due to the light variations detected in the optical by Voloshina & Lyutij (1983) and Voloshina (1986), as well as in the ultraviolet by Lombardi, Giovannelli & Gaudenzi (1987), which are interpreted as ellipsoidal variations of the late-type star or self-occlusions of the hot spot, Echevarría et al. argue that |$i \sim 50^\circ$|⁠, resulting in masses |$M_1 = 0.62$| and |$M_2 = 0.40$| M|$_{\odot }$|⁠. Friend et al. (1990), although they calculate |$K_2 = 155 \pm 3$| km s|$^{-1}$| from the Sodium Na i doublet |$\lambda \lambda$| 8200, 8216 Å, they use the value of |$K_1$| from Hessman et al. (1984) and adopt a value of |$i \sim 41^\circ$| km s|$^{-1}$| derived from q, to calculate the masses of the binary. North et al. (2002) obtained |$K_2 = 165 \pm 1$| km s|$^{-1}$| from the cross-correlation method, and using this value and the rotational velocity |$vsin i$| they calculate |$M_1\sin {i}^3 = 0.36 \pm 0.01$| and |$M_2\sin {i}^3=0.25 \pm 0.01$|⁠. If we use the inclination angle taken from Ritter & Kolb (1998) that they report in their work (⁠|$37 \pm 5$| deg), the mass of the primary is |$M_1 = 1.65 \pm 0.05$|⁠, which is greater than the Chandrasekhar limit. We point out that Ritter & Kolb (2003) have updated the inclination angle estimate to |$51 \pm 5$| deg. If we take this new estimate, the mass of the primary yields |$M_1~=~0.77 \pm 0.02$| M|$_{\odot }$|⁠. This exercise illustrates the strong dependence of the selection on the inclination angle on the final masses of the binary. Bitner et al. (2007) measured |$K_2$|⁠, q, and the fraction of the flux coming from the accretion disc and the WD by fitting spectra obtained on quiescence nights with synthetic spectra for cool stars filling the Roche lobes in close binary systems. Furthermore, by performing an analysis of the ellipsoidal variations in the orbital light curve together with an estimate of the dilution of these variations by the accretion disc flux, they were able to constrain the orbital inclination to the range |$45 < i < 56$|⁠. They obtained |$K_2 = 162.5 \pm 1.0$| km s|$^{-1}$| and |$q = 0.685 \pm 0.015$|⁠. Adopting this values and taking the range of values of i they found mean masses |$M_1=0.81 \pm 0.19$| and |$M_2 = 0.55 \pm 0.13$| M|$_{\odot }$|⁠. The uncertainties in the masses reflect the range of their i values. Finally, Hill et al. (2017) fitted the observed line profiles using Roche tomography to map the surface of the secondary star and derive the system parameters |$\gamma$|⁠, i, |$M_1$|⁠, and |$M_2$|⁠. These parameters were established by matching the observed line profiles to the same |$\chi ^2$| level for various parameter combinations, with the goal of reducing the information content of the reconstructed maps. They found |$\gamma = -15.2$| km s|$^{-1}$|⁠, |$i = 45^{\circ }$|⁠, |$M_1 = 0.94$| M|$_{\odot }$|⁠, |$M_2 = 0.59$| M|$_{\odot }$| (with a mass ratio |$q = M_2/M_1 = 0.628$|⁠) and determined |$K_2 = 163.9$| km s|$^{-1}$| using |$v\sin {i}$|⁠. In addition to this, they measured |$K_2$| and |$\gamma$| using cross-correlation, finding the values |$K_2 = 158.86 \pm 0.14$| km s|$^{-1}$| and |$\gamma = -23.3 \pm 0.1$| km s|$^{-1}$|⁠. This value coincides with our |$K_2$| result within the errors.

The values obtained for the masses shown in Table 4 and discussed above yield values between 0.42 to 1.33 M|$_{\odot }$| for the primary and 0.23 to 0.80 M|$_{\odot }$| for the secondary. These have been obtained by using an inclination angle between 30 and 56 deg. The differences that we see in obtaining the masses of the binary are due not only to the angle i chosen or derived indirectly, but also to the large range of values of |$K_1$| measured by the different authors. On the other hand, the |$K_2$| values do not vary that much, especially the more modern values obtained with a cross-correlation. We point out that our range of values for |$M_1$| is smaller than most publish values. This may be due to our low |$K_1$| value, possibly because our observations of the quiescence nights were very close to the end of an outburst. As we see in Section 4 the disc is not well formed and presents small and high asymmetries. In conclusion, it is evident that the problem of correctly measuring the radial velocity of the emission lines is a core problem that has not yet been resolved. That is, |$K_1 = K_{WD}$| is as unresolved issue. The last direct measurement of |$K_1$| was provided by Echevarría et al. (1989). Since then, it has been indirectly measured only. Added to this, as mentioned above, it is difficult to estimate a reasonable value for the angle of inclination, and only a range of value can be set.

With respect to the |$\gamma$| velocities discrepancies found in this work, we point out that the absortion lines provide a more reliable measurement of |$\gamma$| compared to that obtained from the emission lines (Stover et al. 1980; Hessman et al. 1984; North et al. 2002). This discrepancy arises because emission lines can originate from various parts of the system, such as the accretion disc, disc winds, or hot spots, each with distinct velocity components that may not reflect the true systemic motion.

6 CONCLUSIONS

We presented high-dispersion spectroscopic observations of SS Cyg during the decline stage from an outburst and the beginning of quiescence and determined the orbital parameters of both components at quiescence. This binary is not an eclipsing system and therefore the inclination i is uncertain. Nevertheless, published photometric curves and their model fitting give the range |$45^\circ \le i \le 56^\circ$|⁠. From this interval and our derived mass functions we calculate the following mass ranges: |$0.42\le M_1 \le 0.67$| M|$_{\odot }$| and |$0.19 \le M_2 \le 0.30$| M|$_{\odot }$|⁠, lower than those previously published, and a separation |$1.51 \le a \le 1.77$| R|$_{\odot }$|⁠. We accurately determined the semi-amplitude of the secondary star |$K_2 = 158.6 \pm 0.7$| km s|$^{-1}$|⁠. This precision is attained due to the procedure of constructing one-dimensional spectra, in which cross-correlations could be performed with a large number of absorption lines at the same time. In the case of our semi-amplitude of the primary star |$K_1$|⁠, although lower than previous values, we argue that this may be due to that the accretion disc was still not fully visible and presented large asymmetries. The large scatter found in this parameter in the published previous works and the uncertainty in the inclination angle explain the large range in the inferred masses. Finally, we presented Doppler tomography of several Balmer lines and found that during the decline the accretion disc is barely visible, showing mainly a hot spot, while during the minimum state a disc with an spiral-arm-like structure is observed.

ACKNOWLEDGEMENTS

We thank the daytime and night support staff at the Observatorio Astronómico Nacional at the Sierra San Pedro Mártir for facilitating and helping to obtain our observations. The authors are indebted to DGAPA (Universidad Nacional Autónoma de México) support, PAPIIT projects IN103120 and IN113723. IMZ acknowledges support from CONAHCYT grant 1047189. I want to express my gratitude to all the people who have accompanied me on this journey of my scientific career, which has led me to the publication of my first article. We thank Alejandro Ruelas and Gagik Tovmassian for his help in several issues throughout the paper. We would also like to thank the anonymous referee, whose comments helped improve the content of this article.

DATA AVAILABILITY

The data underlying this article can be shared on request to the corresponding author.

Footnotes

REFERENCES