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A&A
Volume 642, October 2020
Article Number A100
Number of page(s) 14
Section Stellar structure and evolution
DOI https://doi.org/10.1051/0004-6361/201936597
Published online 12 October 2020

© ESO 2020

1. Introduction

Accretion discs are found in a wide range of astrophysical environments such as active galactic nuclei, young stellar objects, and interacting binary stars. Most of the accreting white dwarf (WD) systems known as cataclysmic variables (CVs, Warner 1995) harbour accretion discs, which play a major role in their overall behaviour. The discs in dwarf novae, a subclass of CVs, from time to time undergo outbursts caused by thermal instability, which switches the disc from a low-viscosity to a high-viscosity regime. For a comprehensive review of the disc instability model (DIM), see Lasota (2001). Hameury (2020) reviewed the recent updates of the model.

Short-period (≲2 h) dwarf novae of the SU UMa type show two types of outbursts: normal outbursts lasting a few days, and superoutbursts that have a slightly larger amplitude and a longer duration of a few weeks. The defining property of superoutbursts are superhumps, which are low-amplitude modulations whose period is slightly longer than the orbital period. Superhumps are usually explained by the tidal instability of the accretion disc, which grows when the disc expands beyond the 3:1 resonance radius R3 : 1. This causes the disc to become eccentric and to precess, which initiates superhumps (Osaki 1996). The increased brightness and duration of superoutbursts compared to normal outbursts are explained in the context of the thermal-tidal instability model (TTI), which combines the standard DIM with additional tidal effects such as an enhanced viscous torque that acts when the disc becomes eccentric (Osaki 1989).

It is commonly accepted that the largest disc radius is determined by the tidal influence of the donor star; the viscous and tidal stresses become comparable at rmax and truncate the disc (Paczynski 1977; Papaloizou & Pringle 1977; Ichikawa & Osaki 1994). The 3:1 resonance can only appear in short-period systems where the ratio q ≡ M2/M1 of the masses of donor and WD is small enough, ≲0.25–0.33 (Whitehurst & King 1991). In such CVs, R3 : 1 is lower than rmax. While the responsibility of the tidal instability for the superhump phenomenon is still under debate (Bisikalo et al. 2004; Hameury & Lasota 2005), there is general consensus that during outbursts, the accretion disc expands and then contracts with time (Lasota 2001; Osaki 2005; Hameury & Lasota 2005).

However, in our earlier paper, we showed (Neustroev et al. 2016, hereinafter referred to as Paper I) that the disc radius in the dwarf nova of the SU UMa-type HT Cas has not changed significantly during many years of observations and remained consistently large, close to rmax. Multi-epoch, time-resolved spectroscopic observations from Paper I were obtained between 1986 and 2005 in quiescence, interrupted by several normal outbursts. This result is not consistent with the modern understanding of the evolution of the accretion disc through an outburst cycle, as described above. We also note that Paper I was mostly dedicated to studying the properties of emission structures in the system, of which the dominated source is the extended emission region at the leading side of the disc, opposite to the location of the hotspot from the area of interaction between the gas stream and the disc. This puzzling feature was detected in Doppler maps of many CVs, but its origin still remains unclear. In Paper I we found that the leading side bright spot is always observed at the very edge of the disc.

At the beginning of 2017, HT Cas experienced a very rare superoutburst (the previous superoutburst was observed in 2010, Kato et al. 2012) during which strong superhumps were detected (Enrique de Miguel, vsnet–alert 20570). This event offered us an exceptional opportunity to provide a crucial test for the theory. In the middle of the superoutburst (see Fig. 1), we obtained a new set of time-resolved spectra of HT Cas, which was analysed in a similar manner as our previous observations in quiescence (Paper I). Here we present a comparable analysis of the properties of the accretion disc and of its emission structures during its high and low states. We show the most striking result: the accretion disc radius did not change in the superoutburst because it was already restricted by the tidal limit. Instead, we detected cool gas beyond the Roche lobe of the WD that might have been expelled from the hot accretion disc.

thumbnail Fig. 1.

AAVSO light curve of HT Cas during the 2017 superoutburst (only out-of-eclipse observations are shown).

2. Observations and data reduction

The spectroscopic observations of HT Cas were performed on 2017 January 17 with the Andalucia Faint Object Spectrograph and Camera (ALFOSC) mounted at the 2.5 m Nordic Optical Telescope (NOT) in the Observatorio de Roque de los Muchachos (ORM, La Palma, Spain). The data were obtained under perfect weather conditions with seeing 0.7–0.9″, allowing us to use a narrow slit of 0.5″. The observations were taken with grism 19 in the wavelength range of 4410–6960 Å with a dispersion of 1.2 Å pixel−1 and a corresponding spectral resolution of 2.9 Å. A total of 50 spectra with 120 s individual exposures were obtained, covering one orbital period of the system. He-Ne lamp exposures were taken before, in the middle, and after the observations of the target for wavelength calibration, which is accurate within an rms of 0.040 Å. An accurate wavelength scale for each object spectrum was established through interpolation between the nearest arcs in time. In addition, to extend the wavelength coverage, we also took one spectrum in the wavelength range of 3650–7110 Å using grism 7. Because no spectroscopic standards were observed when the spectra of HT Cas were taken, the latter were first normalised to the continuum (Fig. 2).

thumbnail Fig. 2.

Averaged and continuum-normalised (top panel) and trailed (bottom panel) spectra of HT Cas during its 2017 superoutburst. In the trailed spectrum, displayed twice for clarity, most of the lines consist of a mixture of absorption and emission components. White indicates emission (shown on a linear scale).

About 4.5 h after the end of the NOT observations, we obtained an out-of-eclipse spectrum of HT Cas with the Boller & Chivens spectrograph (B&Ch) attached to the 2.1 m telescope of the Observatorio Astronomico Nacional (OAN SPM) in Mexico (the same instrument as we used in Paper I). In the following night, we obtained another spectrum. The two spectra were flux calibrated with the standard star Feige34, the spectra of which were obtained in the same nights. The continuum of HT Cas in both spectra can be well described by a power law with slightly different indices: −2.85 and −2.60 for the first and second nights, respectively. This difference can be due to either a colour evolution of the object during an outburst (see e.g. Neustroev et al. 2006, 2017) or an orbital colour variation. In the following, we adopt the power index of −2.85 and rescale the NOT spectra to this slope and to match simultaneous optical photometry1. The log of spectroscopic observations is presented in Table 1.

Table 1.

Log of spectroscopic observations of HT Cas.

3. Eclipse ephemerides and orbital phases

Eclipse ephemerides for HT Cas have been derived many times in the past (e.g. Feline et al. 2005; Borges et al. 2008). However, they have accumulated a significant error over time, which resulted in a large discrepancy between the predicted and observed eclipse times. Because the time resolution of our spectroscopy is not sufficient to define an accurate zero-phase, we also used the AAVSO photometric observations of HT Cas during its superoutburst (Kafka 2017). Thus, we extracted 47 times of light minima between JD 2457765 and 2457775. A linear least-squares fit to these times gives the following orbital ephemeris of the light-minima:

HJD min = 245 7765.23997 ( 3 ) + 0.07364720309 · E . $$ \begin{aligned} \mathrm{HJD}_{\min } = 245\,7765.23997(3)+0.07364720309 \cdot E. \end{aligned} $$(1)

In this calculations, we adopted the value of the orbital period Porb = 0.07364720309 d from Feline et al. (2005). The obtained ephemerides were used to calculate the orbital phases of the spectra.

4. Data analysis

The data analysis in this paper is mostly performed in a similar manner to that described in Paper I, and we also used the same plot templates for the figures. Thus, we recommend consulting Paper I for technical details of the analysis and comparing the figures in this work with those of Paper I. Here we also use the same system parameters of HT Cas taken from Horne et al. (1991): M1 = 0.61 ± 0.04 M⊙, M2 = 0.09 ± 0.02 M⊙, q = 0.15 ± 0.03, and i = 81.0 ± 1.0°.

4.1. Averaged and trailed spectra

Figure 2 (top panel) shows the averaged and continuum-normalised out-of-eclipse spectrum, while the measured parameters of the most prominent spectral lines are presented in Table 2. The superoutburst spectrum is quite similar in appearance to those observed in quiescence (see Fig. 2 in Paper I). This is not typical of dwarf novae in outburst; the emission lines during outbursts are usually replaced by broad absorption troughs with weak emission cores (Neustroev et al. 2006, 2017; van Spaandonk et al. 2010; Hernández Santisteban et al. 2019). On the other hand, there are quite many examples of CV outbursts during which the spectra still show strong emission lines (Morales-Rueda & Marsh 2002; Neustroev et al. 2019a). Thus, the case of HT Cas is not unique. We acknowledge the fact, however, that our data cover only a fraction (two days) of the superoutburst during which a complex spectral evolution was possible.

Table 2.

Parameters of the most prominent lines in the mean spectrum.

Despite the similarity of spectra between superoutburst and quiescence, there are at least a few obvious differences. As in quiescence, the outburst spectrum shows double-peaked emission lines of the Balmer series and He I. However, the relative (to the continuum) intensities and the equivalent widths (EW) of these lines are significantly lowered (compare Table 2 and Table 3 from Paper I). The outburst spectrum shows a notable strengthening compared to the Balmer lines of high-excitation emission lines such as He II 4686, C III/N III 4640–4650, and C II 4267. The EW of He II 4686 is comparable in the quiescence and outburst spectra. The double-peaked emission lines of Si II 5669, 6347, and 6371 are clearly detected, although they are rarely observed in CVs. We also observe many absorption lines of Fe II (e.g. λλ4549, 4584, 5169, 5182, 5235, and 5272 Å), and possibly a few of Fe I (λ4518 Å).

In the phase-resolved trailed spectra (Fig. 2, bottom panel), many lines show a mixture of broad emission and narrow absorption components. In the He I and possibly Si II lines, an absorption core between emission peaks goes below the continuum. The iron lines seem to exhibit the absorption core alone, or their emission components are very weak. In the Hα and Hβ lines the absorption component is undetectable in the averaged spectra (the central valley between emission peaks is even more shallow than in quiescence spectra, although it becomes stronger in the higher order Balmer lines), but a sign of absorption is visible in the trailed spectra. The absorption component of different lines varies in phase with each other and with a similar, relatively small radial velocity amplitude. It remains inside the emission peaks and becomes broader at orbital phase ∼0.3, when the lines are blueshifted most strongly. In emission lines, it gives the impression that during these phases, the blue peak is shadowed, becoming weaker (Fig. 3). In mid-eclipse, the absorption components of the Balmer and He I lines almost disappeared or even reversed into emission.

thumbnail Fig. 3.

Trailed spectra (top panel) and averaged profiles (bottom panel) of the Fe II 5169 and He I 5876 lines. The spectra are averaged around the orbital phases 0.3 ± 0.1 and 0.8 ± 0.1. Trailed spectra are shown on a linear scale (white indicates emission).

One of the most important spectral changes between quiescence and the superoutburst is a significant narrowing of the Hα profile in outburst (Fig. 4). It is not only evident from the peak-to-peak separation, which decreased from ∼1100 to 670 km s−1, but also from the full width at half maximum (FWHM), which also decreased from ∼2000 to 1450 km s−1 (Table 2). It is interesting that most of other emission lines also narrowed (the Hβ, Hγ, and Hδ lines are shown in Fig. 4), but they did not become as narrow as the Hα line. In contrast, their width (both the FWHM and peak-to-peak separation) became very similar to each other and close to the Hα width in quiescence. Because the dominant broadening mechanism of double-peaked emission lines is the Doppler shift due to Keplerian rotation of the accretion disc around the accretor (Smak 1969, 1981; Horne & Marsh 1986), the observed narrowing of all emission lines suggests their origin during the outburst from a more extended area than in quiescence. We discuss this in detail in Sects. 6 and 7.

thumbnail Fig. 4.

Averaged profiles of the Balmer lines during the 2017 superoutburst (blue) and in quiescence (2005), shown in red. Hβ, Hγ, and Hδ are also compared with the Hα quiescence profile shown by the dashed black line.

4.2. Light curves

The flux-calibrated spectra were used to compute light curves for the emission lines Hα, Hβ, He Iλ5876, and He IIλ4686. Similarly to Paper I, we computed the light curves by summing the continuum-subtracted flux inside of ±2700 km s−1 window centred at the emission line wavelengths. These light curves are shown in Fig. 5. In the bottom panel of Fig. 5 we show two AAVSO light curves, the phase-averaged one obtained between JD 2457765 and 2457775, and the light curve obtained during our spectroscopic observations.

thumbnail Fig. 5.

Light curves computed for the Hα, He IIλ4686, Hβ, and He Iλ5876 lines (top and middle panels). The bottom panel shows the AAVSO light curves. The black squares represent the phase-averaged data obtained between JD 2457765 and 2457775, and the blue circles show the light curve of these data at the time of our spectroscopic observations.

By comparing the emission line curves with Fig. 3 from Paper I, we conclude that the character of the variability has changed significantly. In superoutburst it has shown single-wave modulations rather than the double wave seen in quiescence. However, the most striking difference is the increase in fluxes (not only in EWs) of some of the emission lines in mid-eclipse. This effect is not seen in high-excitation lines and is hardly visible in Hα, but is clearly detected in Hβ and very strong in the He I lines. It can be explained by assuming that the absorption line components were produced in the innermost parts of the optically thick accretion disc, which are eclipsed by the donor, whereas the emission line region is located more outward in the disc and/or possibly above the orbital plane. Figure 6 (left-hand panel) schematically demonstrates the system geometry in the middle of the eclipse. The inner disc with its absorption spectrum is eclipsed around mid-eclipse phases, while the emission layer can still be seen. In this case, it becomes the main contributor to the spectrum, giving the raise to the total line flux. The more pronounced the absorption component, the stronger the flux increase in mid-eclipse.

thumbnail Fig. 6.

Schematic representation of the HT Cas geometry in the middle of eclipse (left, orbital phase 0.0) and at orbital phase 0.3 (right), when the depression of the blue peak of emission lines is observed.

4.3. Doppler tomography

The Doppler maps of the representative lines are shown in Figs. 7–9. Similarly to Paper I, only out-of-eclipse spectra were used for tomography. In addition to different marks on the maps that facilitate interpreting the tomograms (see the captions to the figures), we also show circles representing velocities at the tidal truncation radius rmax, assuming a circular Keplerian flow in the disc. In Sect. 5 we discuss in detail how significant deviations from circular Keplerian velocities can be, taking into account the gravitational influence of the donor star (see also Fig. 10). For each line we also show in the middle and right panels the trailed spectra and their corresponding reconstructed counterparts. To calculate the map of the absorption line Fe II 5169 (Fig. 9), the latter was first inverted. The complex structure of the He I lines requires negative values in some parts of the Doppler maps to fit the absorption component of the line. To avoid this, we followed an approach proposed by Marsh et al. (1990) that was also applied by Neustroev et al. (2011). Prior to the reconstruction, we have added a Gaussian of FWHM = 600 km s−1 to the data. We then removed the Gaussian equivalent from the calculated tomogram to produce the final result. This procedure has no effect on the goodness of the fit or on the entropy when the latter is determined over small scales.

thumbnail Fig. 7.

Doppler maps and corresponding observed and reconstructed trailed spectra of the Hα, Hβ, and Si II 6347 emission lines. The position of the WD (lower cross), the centre of mass of the binary (middle cross), the Roche lobe of the secondary star (upper bubble with the cross), and the predicted trajectory of the gas stream in the form of the curve are marked. The circle shows the tidal truncation radius rmax of the accretion disc, assuming a circular Keplerian flow. The colour bars indicate normalised flux on a linear scale.

thumbnail Fig. 8.

Doppler maps and corresponding observed and reconstructed trailed spectra of the He I 5876 and 6678 and the He II 4686 emission lines. The position of the WD (lower cross), the centre of mass of the binary (middle cross), the Roche lobe of the secondary star (upper bubble with the cross), and the predicted trajectory of the gas stream in the form of the curve are marked. The circle shows the tidal truncation radius rmax of the accretion disc, assuming a circular Keplerian flow. The colour bars indicate normalised flux on a linear scale.

thumbnail Fig. 9.

Doppler map and corresponding observed and reconstructed trailed spectra of the Fe II 5169 absorption line. The colour bars indicate normalised flux on a linear scale.

The appearance of the tomograms in superoutburst is quite unusual, although some similarity with the Hα Doppler maps in quiescence is apparent (Fig. 11, see also Figs. 6 and 7 in Paper I). No lines show evidence of spiral structures that are sometimes seen in Doppler maps of CVs in outburst (Steeghs et al. 1997). Although a diffuse ring of disc emission is still visible, the Balmer and He I maps are dominated by a bright emission arc at the right side of the tomograms. In Sect. 7 we show that this arc can probably be regarded as an evolved emission region in the leading side of the accretion disc, which we discussed in detail in Paper I. By analogy, hereafter we refer to this structure as the leading arc and to its upper right and lower components as the upper arc and the lower spot, respectively. The leading arc follows the circle representing Keplerian velocities at rmax. In Hα, it is located inside the circle; at its upper part, the arc continues further to the negative x-velocities, where it becomes very bright. This bright, compact, but slightly elongated emission component (in the following referred to as the EEC) starts inside the bubble marking the Roche lobe of the donor star, and moves in the bottom left direction for ∼200 km s−1. The EEC does not follow the predicted trajectory of the gas stream. In Hβ, the ECC is much weaker than in Hα, and it is not visible in other lines. In contrast, the Hβ and especially the He I lines show another bright spot that is consistent with the trajectory of the gas stream (Vx  ≈   − 900, Vy  ≈   + 100). A similar spot has clearly been seen in quiescence, and we identified it as the hotspot that is located well inside the accretion disc (Paper I). Overall, the emission structure in the Balmer and He I lines can be described as having a horseshoe shape with a gap in the third quadrant (−Vx, −Vy). Such a gap was also visible in some Doppler maps of HT Cas in quiescence. The map of the Si II 6347 line looks different than other lines. In addition to the leading arc, which is located at the same position as in the Hβ and He I maps, there is also another arc of a larger radius in the third quadrant.

The He II 4686 emission line is blended with the Bowen blend. This is not a problem for Doppler mapping because a blend does not generate any compact features on the map. However, it can produce a ring-like structure (Marsh et al. 2016). For the case of HT Cas, we can simply ignore this effect because the radius of such a ring on the He II 4686 map should be ≳1700 km s−1, which is almost beyond the edges of the tomogram we showed. He II 4686 is stronger than He I 6678 and as strong as He I 5876, but it does not show a clear double-peaked profile. As a result, its tomogram exhibits a rather diffuse distribution of emission with a compact bright spot in the fourth quadrant (−Vx, +Vy). The latter can be identified as the hotspot located in the region where the gas stream hits the outer edge of the accretion disc. On the other hand, a single-peaked profile suggests non-Keplerian gas motions, for example, perpendicular to the accretion disc. This indicates that at least part of the He II 4686 line is formed in the wind blowing from the disc.

The tomograms of absorption lines (e.g. Fe II 5169) show a compact absorption area, which is associated with the absorption S-wave in the trailed spectra. In the maps it is shifted relative to the position of the WD in the bottom left direction by a few hundred km s−1. A similar absorption pattern is also seen in most of emission line tomograms; it might be responsible for the appearance of the gap in the horseshoe emission structure.

5. Rediscussion of the tidally truncated accretion disc of HT Cas

In Paper I we concluded that the accretion disc radius of HT Cas in its quiescent state is very large, close to the tidal truncation radius rmax. Our conclusion was based on the measured velocity of the outer edge of the disc, Vout, which was accurately measured from the orbit-averaged, double-peaked profiles of the Hα emission line. Assuming circular Keplerian rotation, we determined the radius of the disc to be Rd = 0.52 ± 0.01a, where a is the binary separation. We then compared the obtained Rd with rmax, which was calculated using Eq. (2.61) from Warner (1995), and found that they coincided perfectly. This result is in contrast to the expectations outlined in the introduction and thus raises the question of how reliable the assumptions are on which we base our conclusion. It is known that the gravitational field of the donor star distorts circular particle orbits (Fig. 10, left-hand panel), thus the assumption of the circular Keplerian flow cannot be correct in a strict sense (Paczynski 1977). On the other hand, in Paper I we discussed the expected deviation from a circular Keplerian flow and concluded that although the departures at the outer disc can reach ∼20% from Keplerian velocities (Paczynski 1977; Steeghs & Stehle 1999), the assumption of circular Keplerian velocities should still be reliable for the orbit-averaged spectra because the positive and negative velocity deviations are expected to cancel each other out.

thumbnail Fig. 10.

Left: three representative periodic orbits in the restricted three-body problem calculated for a binary with q = 0.15. The red line shows the largest (truncated) streamline that does not intersect any other orbit. The solid blue and dashed black lines show the orbits whose periods are three and four times shorter than the binary orbital period (3:1 and 4:1 resonance orbits). The Roche lobe is shown with a solid black line. r1, r2, and rmax are different radii of the streamline. Right: truncated, 3:1, and 4:1 orbits shown in velocity coordinates (Doppler map). A and B mark the areas beyond the truncated orbit and their corresponding (roughly) location in the Doppler map. See text for detail.

Bearing in mind the importance of the dynamics of the outer disc regions in reaching a conclusion about the disc radius in HT Cas, we investigated possible deviations from a circular Keplerian flow in more detail. For this, we used solutions of the restricted three-body problem because the tidal influence of the donor on the outer parts of the disc is the primary source of such deviations. We determined the accurate parameters of a few representative periodic orbits in a binary with the system parameters of HT Cas. We note that in the following all the calculated velocities are given for an inclination angle of 81°. We first calculated the largest streamline in the restricted three-body problem that does not intersect any other orbit (we used the code from Ikhsanov et al. 2004). It is generally thought that this streamline represents the limit at which the accretion disc is truncated (Paczynski 1977; Ichikawa & Osaki 1994; Warner 1995). In the following we call it the truncated orbit2. We note that we did not aim to find the last streamline that starts to cross other orbits. Instead, we reconstructed it for the mass ratio q = 0.15 using Table 1 of Paczynski (1977). As expected, the streamline is elongated perpendicular to the line of centres of the two stars (Fig. 10, left-hand panel). The elongation of orbits rises rapidly when it approaches the truncation limit. For comparison, we also calculated and show in Fig. 10 the streamlines whose periods Pstream are three and four times shorter than the binary orbital period Porb (3:1 and 4:1 resonance orbits). For the truncated orbit, Porb/Pstream ≈ 2.81. Following Paczynski (1977), the truncated orbit can be characterised by the radii r1 = 0.434a, r2 = 0.431a, and rmax = 0.522a (Fig. 10, left-hand panel), and the orbit-averaged radius is r ¯ $ {\overline{r}} $ = 0.481a. The particle velocity changes along the streamline, creating an egg-like trajectory in velocity coordinates (Fig. 10, right-hand panel). Assuming that the accretion disc is truncated by this streamline, this results in periodic modulations of the peak-to-peak separation of double-peaked emission lines (Paczynski 1977). The corresponding values of Vout change in the range of 540–670 km s−1 and have the orbit-averaged value of V ¯ out $ {\overline{V}}_{\mathrm{out}} $ = 605 km s−1. V ¯ out $ {\overline{V}}_{\mathrm{out}} $ appears even slightly larger than the circular Keplerian velocity Vmin = 575 km s−1 at rmax, the maximum radius-vector of the orbit3. Thus, the use of such an easily calculated parameter as Vmin allows us to place a solid lower limit on the orbit-averaged V ¯ out $ {\overline{V}}_{\mathrm{out}} $ of the tidally truncated disc. We can conclude now that when the orbit-averaged spectra are discussed, the assumption of the circular Keplerian flow is still reliable even for very large discs. These calculations strengthen the main conclusion of Paper I: the measured V out obs $ V_{\mathrm{out}}^{\mathrm{obs}} $ = 575 ± 4 km s−1 is lower than V ¯ out $ {\overline{V}}_{\mathrm{out}} $ = 605 km s−1, indicating that the disc edge of HT Cas in quiescence is at and even beyond the theoretical truncation limit.

Concluding this section, we note that the question of whether the results from an inviscid calculation can be applied to real discs was studied in detail by Truss (2007). Using hydrodynamic simulations of viscous accretion discs, Truss has shown that the last non-intersecting three-body orbit remains a good estimate of the tidal truncation radius for q >  0.1, and hence for HT Cas.

6. Evidence for the emitting material beyond the truncation limit

As shown above, the deviation from circular Keplerian motion is quite significant in large discs. It might be detectable using time-resolved spectroscopy, for instance through Doppler tomography. In Fig. 11 we show the Hα Doppler map from the 2005 observations of HT Cas in quiescence together with the trajectory of the truncated orbit and for comparison with its circular Keplerian approximation. The emission follows the truncation limit at the left and right sides of the tomogram, but there is a significant excess of low-velocity emission in the upper and especially in the bottom parts of the map (e.g. the lower spot is located about 90 km s−1 beyond the truncation limit).

thumbnail Fig. 11.

Truncated orbit (solid red line), and two non-periodic trajectories (dashed blue and dash-dotted green lines) in the restricted three-body problem calculated for a binary with q = 0.15, shown in spatial (left) and velocity coordinates over the Hα Doppler map from the 2005 observations in quiescence (right). The Doppler map is shown on a linear scale. The dotted red line on the map represents circular Keplerian velocities at rmax.

The exact spatial positions of these areas is difficult to establish. They roughly correspond to the areas A and B in Fig. 10 in which no stable streamlines are possible and so no accurate inverse transformation from velocity to spatial coordinates. However, it can be shown that quasi-circular orbits that extend beyond the truncation limit are consistent with the observed excess of low-velocity emission (Fig. 11, right-hand panel).

The appearance of material outside the truncation limit can have different reasons. In this respect, we must admit that the real shape of accretion discs should be more complex than that of the truncated orbit. First of all, real discs are not inviscid, and viscosity plays a role in the transport of angular momentum and hence in the disc truncation; some deviations from the three-body calculations are not excluded. It was shown that at some physical conditions, the disc can extend beyond the truncated orbit, although only slightly (see e.g. Papaloizou & Pringle 1977). Moreover, the truncated orbit in HT Cas is larger than (although very close to) the 3:1 resonance orbit (Fig. 10, left-hand panel)4, which is believed to be tidally unstable (Whitehurst & King 1991). Smoothed particle hydrodynamics (SPH) models predict that at the 3:1 resonance radius the disc becomes eccentric and precessing (Smith et al. 2007). The SPH calculations show that such a disc has quite a complex shape, and sometimes, ejected streams of matter fill the gap between the tidal orbit and the L1 and L3 Lagrangian points. On the other hand, it has been shown that the tidal instability does not occur in Eulerian models, which use a full energy equation instead of an isothermal approximation (Kornet & Rozyczka 2000, Bisikalo, priv. comm.). It is interesting that the latter models often show the appearance of disc structures consistent with the lower spot in HT Cas (see e.g. Fig. 6 in Kornet & Rozyczka 2000).

It is important to point out that the He I and higher order Balmer lines in outburst follow the predicted truncated orbit very accurately (see the following section). This confirms that the truncated limit adopted in this paper is well justified, supporting the conclusion that there is the emitting material in the quiescent disc of HT Cas beyond the truncation limit.

7. Evidence for the emitting material outside the WD Roche lobe during the superoutburst

The common structure that is clearly seen in the outburst tomograms resembles the most prominent emission area of HT Cas in quiescence: the leading arc. In quiescence, however, this area is most obvious in the Hα line alone, whereas in Hβ it is much weaker, and it is almost undetectable in the He I lines. In superoutburst, in contrast, it is the brightest region in both the Hβ and He I lines, and it is bright in Si II 6347. In Sect. 6 we discussed the spatial location of the leading arc components in quiescence. It is instructive to examine how their location has changed in superoutburst.

Figure 12 compares the location of the leading arc that is visible in Hα in superoutburst and in quiescence (left-hand panel) to that found in He I lines in superoutburst (right-hand panel). First of all, we point out that both components of this emission region in the He I lines (this also holds for Hβ and Si II, see Fig. 7) closely follow the truncation limit, with some extension beyond it. The fact that different emission lines that cover a range of excitation energies and temperatures accurately trace the predicted truncation limit strongly indicates that the accretion disc of HT Cas is indeed truncated and that it is relatively hot (∼15 000 K) until the very edge in superoutbursts.

thumbnail Fig. 12.

Doppler maps combined from the tomograms for the Hα line in superoutburst and quiescence (left) and for Hα and the He I lines in superoutburst (right). The maps are shown on a linear scale. Different colours represent different lines (red and navy show Hα in quiescence and in superoutburst, respectively, and dark red shows the He I lines in superoutburst). The EEC, the brightest emission component in Hα in superoutburst, is flattened to show other emission structures in more detail. The quasi-elliptical dashed lines represent velocities at the tidal truncation limit.

In this respect, it is extremely interesting that during the superoutburst, both the emission components of Hα were shifted toward lower velocities and thus appeared well inside the truncation limit in velocity space (Fig. 12, left-hand panel). The deviation of the upper arc from the limit is at least 100 km s−1, and of the lower spot, it is at least 200 km s−1. The problem is that the upper arc in the Hβ and He I lines is situated in the most elongated area of the truncated orbit, which almost touches the Roche-lobe surface (Fig. 10)5. It leaves no room for low-velocity Hα emission, which could only originate outside the WD Roche lobe. Assuming a Keplerian flow around the centre of mass of a binary, the distance would be at least 1.5 × rmax. We point out, however, that no single orbit in the restricted three-body problem can fully explain the observed emission structures all the way from the lower spot to the upper part of the upper arc. Thus, this is only a rough estimate.

In order to place the Hα emission inside the WD Roche lobe, we need to assume significantly different velocities than predicted in the restricted three-body problem, and that Hα does not follow the dynamics that Hβ and other spectral lines follow. Taking into account the previous discussion and the work by Truss (2007), this appears highly unlikely. The appearance of the Hα emission in the lower velocity region, in contrast with other, hotter lines, can be interpreted as tracing a cooling gas (≲10 000 K) that is expelled from the hot accretion disc through a sector shown in green in Fig. 6.

8. Other emission and absorption components

8.1. Elongated emission component

The cool gas region visible in Hα envelopes the binary for at least 180° in azimuth and visually curves in at the top part of the tomogram to even lower velocities. Nevertheless, it is not clear whether the EEC at the top of the Doppler map and the leading arc are physically linked. One side of the EEC coincides in the Doppler map with the Roche lobe of the donor star. This allows us to speculate about the possible origin of this part of the EEC in the inner semi-sphere of the donor star, which may have been irradiated by the WD and/or hot accretion disc regions. However, although the EEC is the brightest emission area seen in Hα, its other part (a trajectory) cannot be associated with any structures of the binary system such as the gas stream or the hotspot, and none of the hydrodynamical simulations predict an increase in emission in this region. On the other hand, if the EEC is not locked in the rotating binary frame, then its low velocities suggest an origin outside the Roche lobe of the WD. In this respect, we note that if the expelled material leaves the system, then it will create a circumbinary envelope (this resembles the propeller mechanism used to explain Doppler maps of AE Aqr, see Figs. 1–3 in Ikhsanov et al. 2004), the velocity of which are roughly consistent with the EEC. However, the distribution of the material surrounding the system is expected to have an azimuthal symmetry, and we have no physical explanation why we observe only the EEC instead of the ring, and why it is so bright.

8.2. Non-axisymmetrical absorption

The relative compactness of the absorption structure in the Doppler maps suggests that its source is locked in the binary rest frame. Assuming Keplerian velocities, it is located opposite of the donor star, at side of the disc at azimuth ∼110°. To be eclipsed by the donor, the absorption region must be situated close to the WD and it should not be very extended vertically (see Fig. 6, left-hand panel). This resembles the so-called dark spot phenomenon that has been observed in the nova-like UX UMa, which has been explained in the context of the stream-disc overflow model (see Neustroev et al. 2011, and references therein). However, this model cannot explain the depression of the blue peak of the emission lines at orbital phases ∼0.3 ± 0.1, when the absorption component is the most blueshifted (Fig. 3, upper right panel). These two phenomena are probably not physically related.

Such a depression might occur if the very edge of the accretion disc, perpendicular to the line of sight, is temporarily obscured by something in the foreground, for example by a thickened sector of the outer disc (Fig. 6, right-hand panel). This model is consistent with our hypothesis that the bright region on the leading side of the disc in quiescence (and the leading arc in outburst) is caused by irradiation of tidally thickened sectors of the outer disc by the WD and/or hot inner disc regions (see Paper I and Fig. 10 therein). See Sect. 9 for further discussion.

8.3. Hotspot

The canonical hotspot from the area of interaction between the gas stream and the disc is often one of the brightest emission components of dwarf novae in quiescence. This is, however, rarely visible during outbursts, when the hotspot cannot reach a high contrast that would allow it to be detected on the background of the hot and luminous disc. Surprisingly, the hotspot in HT Cas in outburst is easily detected in most of the emission lines (the only exception is Hα). However, only in He II is the spot located at the very edge of the disc, whereas in other lines, it is situated well inside the disc, at distances from the WD of Rhs ≈ 0.23–0.29a. These distances are consistent with those found for the hotspot that is visible in different lines during the quiescent state (Paper I). The difference is that in quiescence, the hotspot velocities were close to Kepler velocities at the spot position (see the left-hand panel of Fig. 11 in Paper I), while during the superoutburst, the velocities were close to the expected velocity of the gas stream at Rhs (Figs. 7 and 8, and the right-hand panel of Fig. 12).

Following Skidmore et al. (2000) and Mason et al. (2000), we explained in Paper I the appearance of the hotspot inside the disc by a very low density of the outer disc regions that allows the gas stream to penetrate deep into the disc. The occurrence of the hotspot at the same location during the superoutburst points to the same explanation, but it also suggests relatively low temperatures in the region of the hotspot origin. The latter is not expected for the disc in the middle of the superoutburst when it is in a quasi-stable high state.

9. Discussion

It is commonly accepted that variations in the outer disc radius in interacting binaries play an important role in understanding the structure and evolution of accretion discs. These variations are predicted by various models of discs (see e.g. Smak 1984; Lasota 2001; Hameury & Lasota 2005), and they are adopted to explain superhumps that are observed during superoutbursts in SU UMa-type dwarf novae (Osaki 2005). Here we mention two important theoretical results obtained by Hameury & Lasota (2005). They found significant variations in the outer disc radius during an outburst (at least 20%), and that the tidal torques that determine the outer radius at which the disc is truncated must be important also well inside rmax. Changes in eclipse widths and eclipse contact times for the bright spot that were detected for several dwarf novae are usually given as observational evidence of disc radius variations (Smak 1996). For instance, this approach allowed Patterson (1981) to deduce that the disc of HT Cas expands by ∼50% during outbursts.

In contrast to these claims, we have shown in Paper I that the disc radius of HT Cas in quiescence was nearly the same during many years of observations. Moreover, we here demonstrated that during the superoutburst, the disc of HT Cas has become hot but remained the same size. This conclusion is independent of the velocity field in the accretion disc and only assumes that this field is globally stable over time. The important empirical fact is that assuming Keplerian rotation, this quasi-constant radius of the disc appears to be close to the tidal truncation limit. This limit in short-period CVs is always larger than the 3:1 resonance radius. This result questions the standard explanation that the appearance of superhumps is a result of the disc radius expansion beyond the 3:1 resonance radius (Osaki 1996).

Admitting that the assumption of the circular Keplerian rotation may not be fully correct, we calculated the last non-intersecting three-body orbit, which was shown to be a good estimate of the tidal truncation limit for the system parameters of HT Cas (Truss 2007). One of our conclusions from these calculations is that the assumption of the circular Keplerian flow is still reliable when the orbit-averaged spectra are analysed. It is even more appealing that when the truncated orbit is visualised in a Doppler map, it traces the leading emission arc even more accurately than for the case of circular orbits. This further confirms our earlier suggestion that the disc of HT Cas is truncated at the tidal truncation limit in both quiescence and outburst.

How does this result depend on the assumption that gravitational forces prevail in the disc? Obviously, if the basic assumption of orderly, gravitational motion breaks down, the emission may not arise where it does according to solutions of the restricted three-body problem. However, we argue that this will not affect our main conclusion that the disc in HT Cas, in both quiescence and outburst, remains near the theoretical maximum radius given by the truncation limit. Observations in quiescence show that the He I and higher order Balmer lines are broader than Hα, differently for each line. This is explained by the fact that they originate in higher temperature areas of the cool disc, closer to the WD. In outburst, however, the width and peak-to-peak separation of most of the lines became very similar to each other and close to those of Hα in quiescence. They also share the same position in the Doppler maps as Hα in quiescence, meaning that the disc is indeed truncated here. Moreover, the fact that different emission lines that cover a range of excitation energies and temperatures accurately trace the predicted truncation limit (Figs. 7–8 and 12) can be interpreted as a sign that the deviations from gravitationally dominated motion of the accretion flows are not very significant6, although we cannot completely exclude them.

The case of HT Cas is not unique. In Paper I we inspected several CVs with relatively well measured system parameters and found that the accretion disc in most of them (VW Hyi, WZ Sge, V406 Vir, EZ Lyn, RR Pic, and IP Peg) also has a radius that is close to the tidal truncation limit. Moreover, it has been reported that in WZ Sge the same disc radius has been observed for 40 years (Mason et al. 2000). All this strengthens our hypothesis that the accretion disc in CVs and possibly in other interacting binaries is always extended to its truncation limit. Much evidence exists that the outermost regions of the disc have quite a low density and much lower optical depth than the inner disc, and they contribute only a tiny fraction to the total optical flux of the system. The “observed” radius variations are thus the variations of the “photometric” disc edge, which can move depending on the physical conditions in the disc. However, beyond this edge, there is still material that repeatedly orbits the WD and mostly contributes to emission lines rather than to broadband continuum light (see Paper I for a further discussion).

We have also presented several lines of evidence that indicate that the Hα emission during the superoutburst most probably originated beyond the Roche lobe of the WD. We propose that we observe here relatively cool material that was ejected from the leading side of the hot disc during the superoutburst. Hα traces cooler gas with lower ionisation than other Balmer and helium lines; this gas in the outflow region can be easily excited by the hot accretion disc to produce the Hα emission line. The amount of expelled matter must be quite significant to be able to obscure the disc edge perpendicular to the line of sight (Fig. 6, right-hand panel). This hypothesis is consistent with predictions of numerical simulations, in which the appearance of the circumbinary envelope created by matter that left the accretion disc and went outside the Roche lobe has been noted (see e.g. Bisikalo et al. 1998). An expansion of material outside the Roche lobe of the accretor was predicted even for systems in the quiescent state. Synthetic Doppler maps of such simulations (Bisikalo et al. 2008) closely resemble the observed tomograms of HT Cas.

This result can have important implication for CV evolution theory, whose predictions are not fully supported by observations of the currently known CV sample7. The evolution of a CV is driven by angular momentum loss from the binary. According to the standard model, angular momentum loss in short-period CVs (below the period gap) is assumed to be driven solely by gravitational radiation (for details, see Knigge et al. 2011). However, the material that leaves the system carries with it the specific orbital angular momentum of the WD. This material is expected to flow out within the orbital plane of the system and thus can create and feed (continuously or occasionally, e.g. during outbursts) a circumbinary disc (CB). It has been shown that the inclusion of angular momentum loss associated with a CB disc can significantly affect the evolution of CVs (Spruit & Taam 2001; Taam & Spruit 2001). In particular, Taam et al. (2003) have shown that even low fractional mass input rates into the CB (δ ∼ 10−4) can promote the mass transfer between the binary components and thus increase the evolution rate and heat the WD by the increased infall of material. Moreover, assuming even lower δ ∼ 10−5, Willems et al. (2005) have demonstrated that this form of angular momentum loss tends to smooth the predicted spike in the number of systems near the period minimum. Deeper discussion of the possible effect of a CB on the CV evolution is beyond the scope of this paper.

How significant and how common is mass outflow from the outer accretion disc in CVs and related objects? Some evidence for a possible expansion of accretion disc material beyond the Roche lobe has been presented for long-period nova-like CVs (see e.g. Hernandez et al. 2017; Subebekova et al. 2020). However, to the best of our knowledge, the observational confirmation of such an expansion in a short-period CV is obtained for the first time. It was possible because HT Cas is an eclipsing binary and its system parameters are known accurately enough, and because it exhibited emission lines during the superoutburst, which is not a common case. Unfortunately, only a few short-period CVs were observed spectroscopically during their superoutbursts, and most of them are non-eclipsed. However, in a few WZ Sge-type dwarf nova exhibiting double-peaked emission lines in superoutburst, the peak-to-peak separation of Hα was found to be significantly smaller than that of other lines (V1838 Aql, Hernández Santisteban et al. 2019, and TCP J21040470+4631129, Teyssier 2019; Neustroev et al. 2019a). Moreover, in TCP J2104070+4631129, all the Balmer and He I lines are much broader in quiescence than they were during the superoutbursts (Neustroev et al. 2019b). We also noted the presence of a horseshoe structure in the Hα Doppler map of SSS J122221.7−311525 (Neustroev et al. 2017, Neustroev et al., in prep.). It is very interesting that this system just after the end of the superoutburst plateau showed the appearance of a strong near-infrared excess resulting in very red colours. The colours then became bluer again, but it took a few hundred days to acquire a stable level. Similar reddening of optical light after the superoutburst has been reported for several other WZ Sge-type stars (see Neustroev et al. 2017, and references therein). All these features probably indicate mass outflow from the outer disc. Unfortunately, the presented observations do not allow for a reasonable estimate of the mass-loss rate through the outer disc. An attempt to estimate this parameter might be made by applying numerical simulations.

10. Summary

We have analysed time-resolved spectroscopic observations of the dwarf nova HT Cas during its 2017 superoutburst with the aim of comparing the properties of the accretion disc in the system during superoutburst and in quiescence. We also discussed again the location of emission structures and the accretion disc size in quiescence using solutions of the restricted three-body problem. The principal results of this study are summarised below.

  1. The superoutburst spectrum is similar in appearance to the quiescent spectra, although the strength of most of the double-peaked emission lines of the Balmer series and He I decreased. In addition, the high-excitation lines of He II 4686, C II 4267, and the Bowen blend significantly strengthened in comparison with the Balmer lines. We also detected the emission lines of Si II, which are rarely observed in CVs.

  2. Many lines show a mixture of broad emission and narrow absorption components. In the He I lines, the absorption extends below the continuum. The iron lines are dominated by the absorption, or their emission component is very weak. In the Hα and Hβ lines, the absorption is undetectable in the mean spectrum, although its sign is visible in the trailed spectra.

  3. Hα in outburst was much narrower than in quiescence. Other emission lines also narrowed in outburst, but they did not become as narrow as Hα.

  4. The single-peaked profile of He II suggests that at least part of this line is formed in the wind blowing from the disc.

  5. Doppler maps of the Balmer and He I lines are dominated by a bright emission arc at the right side of the tomograms, superposed on a diffuse ring of emission. This leading arc is probably an evolved emission region at the leading side of the accretion disc; this is the dominant emission source of the Hα line in quiescence.

  6. In Hα in quiescence, and in the Hβ, He I, and Si II lines in superoutburst, the leading arc is located at and even beyond the theoretical truncation limit, indicating thus that the disc size during the superoutburst remained the same as in quiescence.

  7. Doppler tomography of Hα in superoutburst revealed that the bulk of its emission, which traces cooler gas than other studied lines, is produced beyond the Roche lobe of the WD. We interpret this as a signature that cooling gas is expelled from the hot disc.

These unexpected findings question the standard explanation for superoutbursts and the appearance of superhumps. They can have important implications for CV evolution theory.

1

We used the public data from the AAVSO International Database, http://www.aavso.org/data-download

2

Despite the simplicity of the inviscid three-body formalism that was first used by Paczynski (1977) for calculations of the largest disc size, other approaches were adopted in a number of studies with very similar results (see e.g. Papaloizou & Pringle 1977; Goodman 1993; Truss 2007).

3

rmax is usually calculated using Eq. (2.61) from Warner (1995). This equation approximates the tabulated values of rmax calculated by Paczynski (1977). We found that it is accurate to 1% over the range 0.1 < q < 0.4, but the accuracy declines significantly outside this limit. While it is not a problem for HT Cas, we present here another approximation formula that is accurate to better than 1% over the range 0.03 <  q <  0.73:

r max a = 0.353 + 0.271 e − 3.045 q . $$ \begin{aligned} \frac{r_{\rm max}}{a} = 0.353+0.271\,\mathrm{e}^{-3.045\,q}. \end{aligned} $$(2)

4

The 3:1 resonance orbit is characterized by the following parameters: r1 = 0.417, r2 = 0.428, rmax = 0.492, the orbit averaged radius is r ¯ $ {\overline{r}} $ = 0.459. r ¯ $ {\overline{r}} $ has the same value as that calculated in the usual manner using the formula r3 : 1 = a (1/3)2/3 (1 + q)−1/3. The corresponded values of Vout change in the range of 560–670 km s−1, having the orbit-average value of V ¯ out $ {\overline{V}}_{\mathrm{out}} $ = 617 km s−1.

5

In HT Cas, the equatorial Roche-lobe radius (in the direction perpendicular to the line of centres of the WD and the donor) is 0.558a, and rmax = 0.522a.

6

We also refer to the recent work by Doğan & Nixon (2020). They analysed effects of non-Keplerian rotation in accretion discs and concluded that the departure from Keplerian rotation in even the outer disc regions in CVs is quite small.

7

Here we note the following discrepancies between the available models of CV evolution and the observations relevant to this paper: (1) the measured effective temperatures of WDs in short-period CVs show a large scatter and imply higher mass-transfer rates than predicted (Pala et al. 2017); (2) although the current empirical period distribution reveals a significant accumulation of systems near the observed minimum period (Gänsicke et al. 2009; Kato et al. 2017) that resembles the period-minimum spike predicted by CV population models (Kolb & Baraffe 1999), the width of the empirical spike is larger and the relative number of CVs at the period minimum is much smaller than predicted (Knigge et al. 2011; Pala et al. 2020).

Acknowledgments

We are thankful to the anonymous referees whose comments helped greatly to improve the paper. We acknowledge the financial support from the visitor and mobility program of the Finnish Centre for Astronomy with ESO (FINCA), funded by the Academy of Finland Grant No. 306531. This work was supported by PAPIIT Grant IN-102120. Based on observations made with the Nordic Optical Telescope, operated by the Nordic Optical Telescope Scientific Association at the Observatorio del Roque de los Muchachos, La Palma, Spain, of the Instituto de Astrofisica de Canarias. The data presented here were obtained with ALFOSC, which is provided by the Instituto de Astrofisica de Andalucia (IAA) under a joint agreement with the University of Copenhagen and NOTSA. We acknowledge with thanks the variable star observations from the AAVSO International Database contributed by observers worldwide and used in this research.

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All Tables

Table 1.

Log of spectroscopic observations of HT Cas.

In the text
Table 2.

Parameters of the most prominent lines in the mean spectrum.

In the text

All Figures

thumbnail Fig. 1.

AAVSO light curve of HT Cas during the 2017 superoutburst (only out-of-eclipse observations are shown).
In the text
thumbnail Fig. 2.

Averaged and continuum-normalised (top panel) and trailed (bottom panel) spectra of HT Cas during its 2017 superoutburst. In the trailed spectrum, displayed twice for clarity, most of the lines consist of a mixture of absorption and emission components. White indicates emission (shown on a linear scale).
In the text
thumbnail Fig. 3.

Trailed spectra (top panel) and averaged profiles (bottom panel) of the Fe II 5169 and He I 5876 lines. The spectra are averaged around the orbital phases 0.3 ± 0.1 and 0.8 ± 0.1. Trailed spectra are shown on a linear scale (white indicates emission).
In the text
thumbnail Fig. 4.

Averaged profiles of the Balmer lines during the 2017 superoutburst (blue) and in quiescence (2005), shown in red. Hβ, Hγ, and Hδ are also compared with the Hα quiescence profile shown by the dashed black line.
In the text
thumbnail Fig. 5.

Light curves computed for the Hα, He IIλ4686, Hβ, and He Iλ5876 lines (top and middle panels). The bottom panel shows the AAVSO light curves. The black squares represent the phase-averaged data obtained between JD 2457765 and 2457775, and the blue circles show the light curve of these data at the time of our spectroscopic observations.
In the text
thumbnail Fig. 6.

Schematic representation of the HT Cas geometry in the middle of eclipse (left, orbital phase 0.0) and at orbital phase 0.3 (right), when the depression of the blue peak of emission lines is observed.
In the text
thumbnail Fig. 7.

Doppler maps and corresponding observed and reconstructed trailed spectra of the Hα, Hβ, and Si II 6347 emission lines. The position of the WD (lower cross), the centre of mass of the binary (middle cross), the Roche lobe of the secondary star (upper bubble with the cross), and the predicted trajectory of the gas stream in the form of the curve are marked. The circle shows the tidal truncation radius rmax of the accretion disc, assuming a circular Keplerian flow. The colour bars indicate normalised flux on a linear scale.
In the text
thumbnail Fig. 8.

Doppler maps and corresponding observed and reconstructed trailed spectra of the He I 5876 and 6678 and the He II 4686 emission lines. The position of the WD (lower cross), the centre of mass of the binary (middle cross), the Roche lobe of the secondary star (upper bubble with the cross), and the predicted trajectory of the gas stream in the form of the curve are marked. The circle shows the tidal truncation radius rmax of the accretion disc, assuming a circular Keplerian flow. The colour bars indicate normalised flux on a linear scale.
In the text
thumbnail Fig. 9.

Doppler map and corresponding observed and reconstructed trailed spectra of the Fe II 5169 absorption line. The colour bars indicate normalised flux on a linear scale.
In the text
thumbnail Fig. 10.

Left: three representative periodic orbits in the restricted three-body problem calculated for a binary with q = 0.15. The red line shows the largest (truncated) streamline that does not intersect any other orbit. The solid blue and dashed black lines show the orbits whose periods are three and four times shorter than the binary orbital period (3:1 and 4:1 resonance orbits). The Roche lobe is shown with a solid black line. r1, r2, and rmax are different radii of the streamline. Right: truncated, 3:1, and 4:1 orbits shown in velocity coordinates (Doppler map). A and B mark the areas beyond the truncated orbit and their corresponding (roughly) location in the Doppler map. See text for detail.
In the text
thumbnail Fig. 11.

Truncated orbit (solid red line), and two non-periodic trajectories (dashed blue and dash-dotted green lines) in the restricted three-body problem calculated for a binary with q = 0.15, shown in spatial (left) and velocity coordinates over the Hα Doppler map from the 2005 observations in quiescence (right). The Doppler map is shown on a linear scale. The dotted red line on the map represents circular Keplerian velocities at rmax.
In the text
thumbnail Fig. 12.

Doppler maps combined from the tomograms for the Hα line in superoutburst and quiescence (left) and for Hα and the He I lines in superoutburst (right). The maps are shown on a linear scale. Different colours represent different lines (red and navy show Hα in quiescence and in superoutburst, respectively, and dark red shows the He I lines in superoutburst). The EEC, the brightest emission component in Hα in superoutburst, is flattened to show other emission structures in more detail. The quasi-elliptical dashed lines represent velocities at the tidal truncation limit.
In the text

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