Period Bouncer Cataclysmic Variable EZ Lyn in Quiescence

Cataclysmic variables (CVs) are close binaries that contain a white dwarf (WD) and a late main-sequence star (K-M spectral type) or a brown dwarf. The secondary star fills its Roche lobe and transfers mass to the WD through the inner Lagrange point (L₁). Their orbital period ranges from days to a minimum period of ∼80 minutes. It is assumed that CVs evolve toward shorter orbital periods reaching the period minimum, at which point the secondary star achieves a high level of electron degeneracy (becoming a brown dwarf) and further contraction of the orbit becomes impossible (Paczynski & Sienkiewicz 1981). From that point the orbital period starts to increase and CVs form so-called post-period minimum systems frequently named as period bouncer systems. From CVs' evolution estimations and Galaxy age, it is expected that a number of period bouncer systems should be predominant (Kolb & Baraffe 1999). In the meantime, the mass transfer rate drops making period bouncer systems less luminous and difficult to find. Nevertheless, the number of candidates for such systems significantly increased in the last decade (Kato et al. 2009b).

One of the bona fide period bouncers is EZ Lyn (SDSS J080434.20+510349.2 or SDSS0804), identified as a CV by Southworth et al. (2006). Pavlenko et al. (2007) found this object in a WZ Sge-type superoutburst on 2006 March 4 reaching V_max = 12.8. In addition to a large amplitude superoutburst it underwent rebrightening (Pavlenko et al. 2007; Shears et al. 2007; Kato et al. 2009a) during the decline, a characteristic proper to WZ Sge objects. EZ Lyn has a 0.0590048 day orbital period (Zharikov et al. 2008; Kato et al. 2009b; Pavlenko 2009), and a superhump period of 0.060 day (Pavlenko et al. 2007, 2011). The mass ratio q ≡ M₂/M_WD = 0.056 (Zharikov et al. 2008; Isogai et al. 2015) was estimated from the empirical relation between the superhump period excess = 0.011 (Kato et al. 2012) and the mass ratio (Patterson et al. 2005a).

Pavlenko (2007) discovered WD non-radial pulsations that appeared 8 months after the 2006 superoutburst, and lasted about 2 yr. In 2006 December–2007 January a series of mini-flares were observed by Zharikov et al. (2008). In 2007–2008 the light curve of EZ Lyn showed sinusoidal variability, after the object's magnitude decreased to V ≈ 17.7(1), ⁶ with half of the orbital period (42.48 minutes) and an ≈0.07 mag amplitude. A possible presence of grazing eclipses during the 2006 superoutburst (Kato et al. 2009b), suggests a high-inclination system.

In 2010, only 4 yr after the previous, the object showed another unexpected superoutburst. It returned to the quiescent level by the beginning of 2012. In 2012, the object again showed a double-hump light curve, but with a significantly smaller (∼0.01–0.02 mag) amplitude. Based on time-resolved spectroscopy and photometry obtained in 2012, Zharikov et al. (2013) proposed that the system contains a massive M_WD ≳ 0.7 M_☉ WD with T_WD ∼ 12,000 K temperature, a late-type brown dwarf, a large accretion disk with two outer annuli spirals caused by the 2:1 resonance, and an optically thin accretion disk. From the light-curve simulation Zharikov et al. (2013) conclude that the system inclinations is about ∼75°–80°. Pavlenko et al. (2014) showed the existence of a stable signal around 100 cycles per day (c d⁻¹) and three signals around 310, 338, and 368 c d⁻¹ (the corresponding periods are 864, 279, 256, and 235 s). They interpreted them as independent non-radial pulsations of the WD. The average brightness of the object in their study was about V ≈ 17.7 (see Figure 1 where we present the entire recorded light curve of EZ Lyn, including American Association of Variable Star Observers (AAVSO) data accumulated up to 2020).

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In this paper, we report new observations of EZ Lyn in quiescence, obtained about 9 yr after the last superoutburst, to probe the accretion disk conditions.

The Gaia parallax of EZ Lyn is π = 7.00(11) mas, which corresponds to a distance of D = 142.8(2.8) pc (Green et al. 2019a). The total Galactic extinction in this direction is E(B − V) = 0.0475(22) (Schlafly & Finkbeiner 2011). It is probably less than E(B − V) = 0.01 (Green et al. 2019a) to EZ Lyn located in our close vicinity.

This study aims to define the system's fundamental parameters and try to probe the state of accretion disk structure in quiescent. In Section 2, we describe our observations and the data reduction. The results of photometry and spectroscopy are described in Sections 3–5. The parameters of the WD and system fundamental parameters are estimated in Section 6. The light-curve modeling is presented in Section 7. The structure of the accretion disk from the emission-line profile modeling and Doppler tomography are discussed in Section 8. The summary of the obtained results and their application to period bouncer systems is given in Section 9.

New spectroscopic observations of EZ Lyn were performed during two nights on 2018 October 30 and November 1, with the Andalucia Faint Object Spectrograph and Camera mounted at the 2.5 m Nordic Optical Telescope (NOT) in the Observatorio de Roque de los Muchachos (La Palma, Spain). The weather was clear on both nights with a variable seeing of 1.0''–1.5''. On the first night, the grism #19 and a slit width of 1.3'' were used, yielding a spectral resolution of R ∼ 750 in the spectral range 4410–6960 Å. While on the second night of observations, the data obtained with the grism #7 and a slit width of 1.0'' provided a spectral resolution of R ∼ 650 in the spectral range 3650–7110 Å. ⁷ A total of 16 and 19 spectra were collected on the first and the second night, respectively. The time of an individual exposure was 300 s. Approximately a full orbital period of the system was covered each night (see Table 1 for detail). A He–Ne lamp was exposed before and after the target for wavelength calibration. The spectrophotometric standards HD 84937 and G191-B2B were used for the flux calibration. The obtained images were bias corrected and flat fielded. After that, the spectral data were reduced in the standard way using apextract and onedspec IRAF ⁸ tools. The fluxes were verified using multicolor photometric observations on 2018 October 30 at the same telescope and instrument. These single-shot exposures were taken through Bessel BVRI and Sloan Digital Sky Survey (SDSS) z filters. Besides, we also got near-infrared (NIR) observations with the JHK_s filters. They were performed on 2018 November 2 with the NOTcam instrument on the NOT. Throughout the paper, this data set is referred to as not-58423, where 58423 stands for the average epoch in MJD of those observations.

Time-resolved photometry of EZ Lyn was obtained in the V band using the direct CCD image mode at the 0.84 m telescope of the Observatorio Astronómico Nacional at San Pedro Mártir (OAN SPM) in Mexico and in the clear CCD APOGEE E47-MB detector mode at the Shajn 2.6 m telescope of the Crimean Astrophysical Observatory (CrAO). The maximum of sensitivity of the APOGEE E47-MB detector is close to the central wavelength of the V band. The images were corrected for bias and flat fielded. The field star with V = 16.04 from Zharikov et al. (2008) (α₂₀₀₀ = 08^h04^m39.7ˢ, δ₂₀₀₀ = 51°04'50.1'') was used for the magnitude calibration. Photometric data of OAN SPM were calibrated using Landolt standard stars. The photometric errors were calculated from the dispersion of the magnitude of comparison stars in the object field. They range from 0.01–0.03 mag. A log of photometric observations is given in Table 1.

On 2019 August 24 we performed 3.3 ks observation of EZ Lyn with the Neil Gehrels Swift Observatory (Gehrels et al. 2004), using both the X-ray Telescope (XRT; Burrows et al. 2005) and the UV/Optical Telescope (UVOT; Roming et al. 2005). The data were processed and analyzed using heasoft 6.28, together with the most recent version of the calibration files.

In this paper, we also make use of five spectra of EZ Lyn obtained during the course of SDSS. These spectra are referred to as sdss-53090, sdss-55517, sdss-55542, sdss-55559, and sdss-55590. In addition, we add the spectra obtained in 2008, 2009, and 2012 with the Boller & Chivens spectrograph and the 2.1 m telescope of OAN SPM. The description of those observations, denoted in the paper as spm-54503, spm-54537, spm-54857, and spm-55952, can be found in Zharikov et al. (2013).

Figure 2 presents light curves obtained in 2019 and 2020 (see Table 1) folded with the orbital period of the system. The average magnitude of the object was V = 17.95(10). The object was slightly fainter than it was on 2003 October 24 (V ≈ 17.85) before the first superoutburst (see SDSS DR15 data). The source shows some brightness variations around the average value with a maximal amplitude of about 0.1 mag. This variability is significantly higher than the error of measurements (see Figures 2, 3). The fast Fourier transform analysis of 2019 data shows only wide and weak peaks in the power spectrum at frequencies that correspond to the full and half-orbital periods. In 2020, in addition to these two, 12.5 minute pulsations were clearly detected. The power spectrum in Figure 4 shows significant peaks at the orbital, half of the orbital periods, and the 12.5 minute pulsation along with its 2nd harmonics. An important outcome of the last observations is confirmation of grazing eclipses in the light curve. Moreover, the latest observations of 2021 March feature a long-term quasiperiodic variability commonly detected in SDSS J123813.73-033933.0 (Zharikov et al. 2006; Aviles et al. 2010; Pala et al. 2019) (see the bottom panel of Figure 3). Such behavior was registered in EZ Lyn light curves before the 2006 superoutburst (Szkody et al. 2006).

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A multiband photometry in BVRI of the object was obtained simultaneously with our spectroscopic observations in 2018 October 31 (JD = 2458422.0) using NOT. The magnitudes of the object were B = 18.06(3), V = 17.95(3), R_c = 17.75(3), and I = 18.11(5), where errors are intrinsic uncertainties of the measurement. In 2019 February, the object was observed in the UBR_c I_c bands. The measured magnitudes in the bands were U = 17.86(10), B = 18.09(10) R_c = 17.90(10), and I_c = 17.95 (10) for JD = 2458526. Here, the deviations from the average magnitudes are conditioned by the flickering, rather than precision of the photometry. Orbital variability is very weak, if present. The absolute magnitude of the object from the Gaia parallax is M_V = 12.05(11), where the error includes uncertainties of the magnitude, the distance, and the interstellar absorption.

In 2018 March (JD = 2458208.0), EZ Lyn was also observed in the infrared bands. The calibration of NIR data was obtained using five field Two Micron All Sky Survey stars with magnitudes in the 14.0–15.0 range. The object JHK magnitudes are J = 17.84(3), H = 17.47(5), and K = 16.91(3). These magnitudes are significantly fainter than J = 17.29(5), H = 16.97(5), and K = 16.41(6) (V ≈ 17.1; 2007 March) reported by Kato et al. (2009b) and J = 17.3(1) (V ≈ 17.7; 2009 October) by Zharikov et al. (2013). This means that the contribution of the accretion disk in NIR flux was significantly higher in the previous observations. Based on the latest data, the M_K absolute magnitude of the secondary is M_K ≳ 11.1 confirming the brown dwarf identification of the secondary with spectral class L2 or later (Faherty et al. 2012).

Swift-XRT detected a very weak X-ray source with a count rate of ${1.6}_{-0.5}^{+0.7}\times {10}^{-3}$ count s⁻¹ at the object position. Because only four source counts were detected, no meaningful spectral analysis is possible. Nevertheless, assuming that the spectrum of EZ Lyn is similar to other WZ Sge-type stars, such as SSS J122221.7-311525 and GW Lib (Neustroev et al. 2018), and using the count rate as the scale factor, we estimated the unabsorbed X-ray flux of EZ Lyn in the 0.3–10 keV energy range to be 5.9 × 10⁻¹⁴ erg s⁻¹ cm⁻². Adopting the distance of 142.8 pc, this corresponds to the unabsorbed X-ray luminosity of 1.4 × 10²⁹ erg s⁻¹. Such luminosity is in agreement with those found for other WZ Sge-type stars in quiescence and accreting WDs (Reis et al. 2013; Neustroev et al. 2018).

At the moment of Swift observations (2019 August 24) the UV magnitudes of EZ Lyn were uvw2 = 16.92(4), uvm2 = 16.80(7), uvw1 = 16.85(5), u = 17.03(5), b = 18.04(7), and v = 17.99(15). The fluxes were dereddened using the appropriate value of E(B − V) = 0.01 taken from Schlegel et al. (1998) and Schlafly & Finkbeiner (2011), with A_λ /E(B − V) ratios calculated for UVOT filters using the mean Galactic interstellar extinction curve from Cardelli et al. (1989).

All spectra (see Table 2) that we analyze in this paper were obtained in quiescence, either before the first superoutburst, between superoutbursts, or after the second superoutburst (see Figure 1). The spectra are dominated by strong and broad absorption lines from the WD flanking double-peaked Balmer emission lines from the accretion disk (see Figure 5, top panel). Although all spectra are similar in appearance, they exhibit a notable difference in the strength of emission lines, especially of the higher-order Balmer lines. Perhaps the most prominent example of such variations comes from the SDSS observations. Four of five SDSS spectra were obtained in 2010 during a 73 day time interval when the optical flux of EZ Lyn did not change significantly (Isogai et al. 2015). However, the emission lines in these spectra show prominent variations in strength in a non-monotonic fashion (Figure 1, bottom panels, and Table 2).

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Table 2. The Parameters of the WD and the Accretion Disk Contribution from the Spectra Fitting

Obs. Data	S/N	Teff (K)	Fd/Fsys	${M}_{V}^{\mathrm{disk}}$	Flux (erg s−1 cm−2)			B.D.	Model Parameters
+2400000	(4600 Å)	($\mathrm{log}g=8.35$)	(%)	(mag.)	Hα	Hβ	Hγ	Hα/Hβ	Vout (km s−1)	b	rin/rout
sdss-53090	25	11,500	13	14.30	1.7 × 10−14	1.1 × 10−14	7.0 × 10−15	1.63	650	1.26	0.15
spm-54503	60	14,750	7	14.55	9.1 × 10−15	6.2 × 10−15	4.6 × 10−15	1.47	679	0.47	0.06
spm-54537	60	14,750	4	15.34	7.7 × 10−15	5.1 × 10−15	4.6 × 10−15	1.50	676	0.54	0.07
spm-54857	40	13,750	4	15.28	7.5 × 10−15	5.4 × 10−15	4.3 × 10−15	1.38	663	0.51	0.05
sdss-55517	50	14,250	13	13.92	8.2 × 10−15	5.6 × 10−15	5.1 × 10−15	1.48	616	1.21	0.03
sdss-55542	50	14,000	14	13.77	1.1 × 10−14	5.4 × 10−15	4.8 × 10−15	1.95	651	0.96	0.05
sdss-55559	40	14,500	6	14.76	1.3 × 10−14	8.4 × 10−15	6.6 × 10−15	1.56	652	1.05	0.04
sdss-55590	40	14,250	3	15.48	1.7 × 10−14	9.5 × 10−15	6.5 × 10−15	1.77	644	0.85	0.05
spm-55952	50	13,000	2	16.42	1.3 × 10−14	7.6 × 10−15	4.5 × 10−15	1.70	698	0.52	0.06
not-58423	30	11,250	13	14.35	8.2 × 10−15	5.0 × 10−15	3.1 × 10−15	1.64	686	0.50	0.07

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The SDSS spectra, except for sdss-53090, show the presence of the Mg ii λ4481 Å absorption line. The line is also visible in the individual spectra of the not-58423 data set, and is marginally detectable in the averaged spm-55952 spectrum. Figure 6 shows portions of the sdss-55590 and not-58423 spectra around the feature. Table 3 presents the measured velocities and equivalent widths (EWs) of Mg ii 4481 in the averaged spectra. The Mg ii absorption line originates in the photosphere of the accreting WD and thus is subject to gravitational redshift, allowing a measurement of the WD mass directly (see, e.g., van Spaandonk et al. 2010, and references therein), and also the radial velocity semiamplitude of the WD K₁. Although the Mg ii line is visible practically in all orbital phases of the not-58423 set (Figure 6), a low spectral resolution and a low signal-to-noise ratio (S/N) of individual spectra prevent us from obtaining realistic measurements of the radial velocities.

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Gravitational redshift velocity is a function both of mass and radius (Greenstein & Trimble 1967):

$$\begin{eqnarray}&&{\upsilon }_{\mathrm{grav}}=\displaystyle \frac{{GM}}{{cR}}=0.637\,\displaystyle \frac{{M}_{\mathrm{WD}}}{{M}_{\odot }}\displaystyle \frac{{R}_{\odot }}{{R}_{\mathrm{WD}}}\,\mathrm{km}\,{{\rm{s}}}^{-1},\end{eqnarray} \tag{ 1 }$$

where G is the gravitational constant, c is the speed of light, and R_WD and M_WD are the radius and the mass of the WD, respectively. The observed Mg ii velocity υ_obs = + 49(20) km s⁻¹ is a sum of υ_grav and the systemic velocity γ: υ_obs = υ_grav + γ. Thus, to determine υ_grav one needs to know γ, which is most accurately measured from radial velocity variations of the donor star (the contribution of the low mass secondary to the redshift is negligible). Unfortunately, the spectral lines of the donor are not detected in EZ Lyn. However, we can reasonably assume that γ is close to zero. This assumption is based on the small distance to the object and the Galactic latitude (b = +32.0°). If this is correct, then using the WD mass–radius relationship from Hamada & Salpeter (1961) we get ${M}_{\mathrm{WD}}={0.80}_{-0.21}^{+0.15}$ M_☉. This value appears to be very consistent with other estimates (see below), although obviously the γ velocity must be accurately measured to confirm this result.

The Balmer absorption lines of EZ Lyn provide insights into the properties of the WD and can be used to determine its temperature and surface gravity (T_eff and $\mathrm{log}g$). An estimate of $\mathrm{log}g$ and T_eff of an isolated WD is often done through a spectral fit of a grid of WD atmosphere models to the observed absorption lines. In the case of accreting WDs, the situation becomes more complex due to the presence of broad emission lines and of an additional continuum component, both emanated by the accretion disk. There is also an input from the donor star, but in low mass transfer rate CVs, such as EZ Lyn, this contribution is negligible at short wavelengths (≲5000 Å). The disk emission lines are narrower than the WD absorption troughs, hence they can be removed from the spectrum. But accounting for the disk continuum is a more complex problem because its shape is still unknown. It is assumed that the disk spectrum is to follow a power law, but there are indications that in the low mass transfer rate regime the disk emission can be better approximated by a hydrogen slab model (see, e.g., Pala et al. 2018, 2019). However, we use a power law for continuum F_disk = C λ^−γ , where C and γ are the scaling factor and the power-law index. Our simplified approach is justified because we fit only a portion of spectrum between Hβ and H, covering a relatively short wavelength range.

Our fitting technique can be described as follows. The first step is to cut off the Balmer emission lines which can extend up to a few ×1000 km s⁻¹. The maximum possible extension of the emission-line wings is determined by the radius of the WD and the orbital inclination. We found that for EZ Lyn the optimal cutoff limit ΔV is ∼3500 km s⁻¹. Because the width of higher-order Balmer absorption lines of the WD (blueward of H) becomes comparable with that of the emission disk component, we include in our analysis only the Hβ, Hγ, and Hδ lines and the red wing of H (the Ca ii 3933 Å emission line affects the blue wing of H). We then perform the χ² fit

$$\begin{eqnarray}&&{\chi }^{2}=\sum \,{\left(\displaystyle \frac{{F}_{\mathrm{obs}}-{F}_{\mathrm{calc}}}{{\sigma }_{\mathrm{obs}}}\right)}^{2}\end{eqnarray} \tag{ 2 }$$

of an object spectrum to a grid of theoretical models of DA WDs, to which the disk flux was added

$$\begin{eqnarray}&&{F}_{\mathrm{calc}}={C}_{\mathrm{WD}}{F}_{\mathrm{WD},\mathrm{model}}+{F}_{\mathrm{disk}},\end{eqnarray} \tag{ 3 }$$

where C_WD is the scaling factor of WD spectrum.

We used a grid of LTE pure-hydrogen models from Detlev Koester (Koester 2010), which we retrieved from the Theoretical Spectra Web Server. ⁹ The models cover T_eff = 9000–40,000 K in steps of 250 K, and log g = 6.5–9.5 in steps of 0.25. All WD models were convolved with the appropriate Gaussian instrumental profile to match the spectral resolution of observed spectra. Following the described procedure, we performed a χ² fit to each flux-calibrated and dereddened spectrum of EZ Lyn. As expected, the calculations complied with a range of $\mathrm{log}g$ values from 8.25–8.75. Weighting each spectrum according to its S/N in the continuum at λ = 4600 Å, we obtained the average $\mathrm{log}g=8.39(15)$. The best $\mathrm{log}g=8.35(11)$ value follows from the combination of constraints imposed by the WD radius relation to the $\mathrm{log}g$ (υ_grav = 49(20) km s⁻¹) and the mass–radius relation for different composition WDs (see Figure 7). According to Salpeter (1961), Nauenberg (1972), and Carvalho et al. (2018) the radius of the WD is ${R}_{\mathrm{WD}}={6580}_{-1400}^{+1600}$ km, and the mass is in the range of 0.75 ± 0.25 M_☉, depending on its interior composition. Nonzero temperature models of WDs do not change results significantly (Romero et al. 2019, see Figure 13 for example).

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Fixing the surface gravity of the WD at $\mathrm{log}g=8.35$, we repeated the χ² fit to all 10 spectra again. The values of the χ² were close to one per degree of freedom in all cases. Estimated WD temperatures are listed in Table 2 and are shown Figure 1 (middle panel). We note that for four SDSS spectra, which were obtained in 2010 in a short time interval, the fit gives nearly the same T_eff of 14,250(250) K but a different contribution of the accretion disk produces a quite different spectral appearance. Similarly, T_eff from the two SPM spectra (spm-54503 and spm-54537) obtained in a month's time interval is found to be the same. Meanwhile, the fit of the effective temperature in spm-55952 spectrum is consistent with the WD temperature estimate based on UV Hubble Space Telescope Cosmic Origins Spectrograph (HST-COS) observations (HJD 2455868), obtained only about 3 months earlier (Szkody et al. 2013b). Overall, the WD temperature can be described as being higher (13,000–15,000 K) just after and between the superoutbursts due to compressional heating of the increased infall of accreted material, and much lower (∼11,000 K) after a long period of quiescence (see Table 2 and Figure 1). This is in general agreement with observations of other dwarf novae (see Pala et al. 2017, and references therein).

Assuming the mass ratio q = 0.056 the mass of the secondary is M₂ = 0.028–0.056 M_☉ ≈ 42(14) M_J mass of Jupiter. The separation of the stellar components is a = 0.59 R_☉ = 4.1(9) × 10¹⁰ cm and the secondary's radius is R₂ = 0.103 R_☉ = 1.02R_J. The tidal limitation radius ¹⁰ of the disk is ${R}_{\mathrm{disk},\max }=0.34\ {R}_{\odot }$ and the minimal possible Keplerian velocity in the disk is υ_disk,out = 650 km s⁻¹.

The UV-optical-IR spectral energy distribution (Figure 8) of EZ Lyn is based on the last Swift and NOT observation. It confirms a negligible contribution of the disk in the continuum. An IR excess in the H and K_s bands is provided by the contribution of the L2 brown dwarf at the Gaia distance. The flux in Swift UV data slightly exceeds the expected emission from the WD, and is more probably related to the accretion disk.

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It was noted in Section 1 that the light curves of EZ Lyn obtained during 2006 and after the second 2010 superoutburst showed the presence of barely perceptible eclipses (Kato et al. 2009b; Szkody et al. 2013b; Pavlenko et al. 2014). The eclipses were also visible in our most recent 2020 observations (Figures 2 and 3). The eclipse is best demonstrated in a phase-folded light curve obtained by Pavlenko et al. (2014) in 2013 January and plotted in the top panel of Figure 9 by black points. We applied the binary code developed by Zharikov et al. (2013) to model this light curve. Notably, the object then was about ∼20% brighter (at V ≈ 17.7) than during our last observations (Figure 1). The depth of the eclipse was about ∼0.05 mag. Naturally, such shallow eclipses are not produced by the total occultation of the WD. In the best case, it could be a partial eclipse of the WD, which limits the system's inclination to i ≲ 80°. However, the full width of the observed eclipse at half of its depth is about Δϕ ≈ 0.06 of the orbital phase, but the partial eclipse of the WD with the similar depth gives only the width of Δϕ ≈ 0.005. The amplitude of the humps in the light curve was about of ≈0.02 mag with a slight difference between them.

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Szkody et al. (2013b), from the absence of half-orbital period variability in UV HST-COS data, concluded that the optical double-humped light curve is not related to the WD or its close vicinity. Zharikov et al. (2013) proposed that the double-humped light curves of period bouncer candidates observed in the continuum can be reproduced by a ring-like disk with spiral wave structures. The outer ring-like disk undergoes tidal compression as a result of 2:1 resonance at the corresponding radius, forms the spiral pattern, and contributes to both the emission lines and the continuum. There are some hydrodynamic models of the accretion disk in short orbital period systems (Kunze & Speith 2005; Truss 2007; Lukin et al. 2017), which predict spiral-like structures in the accretion disk and some extended disk shape at the disk's part opposite to the hot spot.

We successfully reproduced the double-humped and eclipse light curve of EZ Lyn (see Figure 9). The light curve was fitted by a model that includes white and brown dwarf stars, a stream of accretion matter, a thin ring-like accretion disk with spiral-like patterns, and two extended spots(arcs) at the outer edge of the disk (Figure 10). The WD is a sphere, defined by the mass–radii relation (2.83b) in Warner (1995a). The secondary is assumed to fill its Roche lobe, and the Roche lobe shape is directly calculated using Equation (2.2) in Warner (1995a). The surface of each component of the system is divided into a series of triangles. We assume that each triangle emits as a blackbody with the corresponding temperature. Each element's intensity is convolved with the corresponding filter bandpass and converted into the flux, taking into account the element surface, orientation, distance to the system, and interstellar absorption. The light curves of individual components and the binary system were obtained by integrating the emission from all the elements lying in view.

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We have selected the input system parameters close to the values described in Section 6. The free parameters of the fit were the mass M_WD of the WD and its temperature T_WD, the system inclination i, the mass transfer rate $\dot{M}$, the temperature of the secondary T₂, the inner R_disk,in and outer R_disk,out radii of the accretion disk, the disk height at the outer radius h_disk,out, and the parameters of hot spots and the spiral arms. The latter were presented in the original paper of Hachisu et al. (2004). The first hot spot is characterized by a length (in degrees), a width (in percent of the accretion disk outer radius), and a temperature excess γ_spot (in percent from the disk temperature). It is located at the impact region between the stream and the outer rim. The shape of the hot spot is described by the schematic model with nonuniform temperature distribution gradient. It is hotter near the impact point and declines in temperature according to an arbitrarily selected manner described by the following equation:

$$\begin{eqnarray}&&{T}_{s}(\varphi )={T}_{d}(1+{\gamma }_{\mathrm{spot}}\times f(\varphi )),\end{eqnarray} \tag{ 4 }$$

where φ is the angle between the line connecting the stars in conjunction and the direction of the spot viewed from the center of mass; f(φ) ≡ a × φ + b is a linear function with constants a and b defined by boundary conditions of f = 1 at the maximum of the hot spot temperature and f = 0 at ends of the hot spot, γ_spot is a free parameter, and T_d is the effective temperature ¹¹ of a steady-state disk. Disk/stream impact points are also free parameters. In addition to the usual hot spot, we need to include an extra element (a second spot) located at the opposite side of the disk to the disk-stream impact hot spot, to achieve a better fit. The presence of the second spot reflects the noncircular shape of the accretion disk. It is characterized by the length (in degrees), the position (in degrees), and the temperature excess γ. In the model the zone has a fixed width filling space between the disk's outer rim and the boundary of the primary Roche lobe. The position angles of hot spots are measured from the impact point between the stream and the disk.

The spiral pattern, following Hachisu et al. (2004), can be described by multiplying thickness of the disk h_disk with z_h , as defined below:

$$\begin{eqnarray}&&{z}_{h}=\max \left(1,\displaystyle \frac{\xi }{(\sqrt{\left(r/{R}_{\mathrm{disk},\mathrm{out}}\right)-{\exp }\left(-\eta \left(\phi -\delta -{i}_{s}\right)\right){\left.\right)}^{2}+{\epsilon }^{2}}}\right),\,\end{eqnarray} \tag{ 5 }$$

where η is the inverse of pitch angle, δ—position angle of the spirals against the binary components, is the width of the spiral pattern, i_s = 0° for the first spiral and 180° for the second, r and ϕ the disk element coordinates in the polar system. In addition to the original model we put an excess of the effective temperature γ_spiral in the spiral (z_h > 1) defined as

$$\begin{eqnarray}&&T(r,\phi )={T}_{d}(1+{\gamma }_{\mathrm{spiral}}\times {z}_{h}(r,\phi )).\end{eqnarray} \tag{ 6 }$$

The gradient descent method was used to find the minimum of the χ², which is an analog of Equation (2) where fluxes are replaced by magnitudes and their errors. The error of each parameter was calculated by the Gaussian fitting of the χ² function when other parameters are fixed at the best values. The best fit model light curve is shown (the red line) in Figure 9. The model parameters are given in Table 4 and the configuration of the system is plotted in Figure 10.

Table 4. System Parameters from in the Light-curve Modeling

Fixed:
Porb	5079.60 s	E(B − V)	0.01
Distance	142.8 pc	q	0.056
Variable and their best values:
i		79.0° (2)
MWD		0.85 (1) M☉
TWD		11248 (40) K
$\dot{M}$		2.7 (1) × 10−12 M☉ yr−1
T2		$\leqslant {1900}_{-1000}^{+400}$ K
Parameters a of the disk:
Rdisk,in		0.203 (2) R☉
Rdisk,out		0.349 (18) R☉
hdisk,out		0.002 (1) R☉
Parameters a of spirals:
ξ—the amplitude of a spiral		0.413 (2)
η—the inverse of the pitch angle
of logarithmic spiral		0.380 (5)
δ—position angle of the spirals
against the binary components		41.3° (4.5°)
—the width of the spiral pattern		0.059 (3)
γ—Temp. excess of spirals		31 (4)%
Parameters of hot spot/lines:
Length spot(1)		89.6° (25.5°)
Width spot (1)		5.0 (2.5)%
γ—Temp. excess spot(1)		10 (4)%
Shift spot (1)		−20.0°${}_{-5}^{+10}$
Shift Tmax spot(1)		−7 (6)
Length spot(2)		53.0° (7.0°)
Width spot(2) fixed		17%
γ—Temp. excess spot(2)		1${}_{-1}^{+2}$%
Shift spot(2)		152.5° (7.5°)
Calculated:
a		0.59 R☉
M2		0.048 M☉
R2		0.11 R☉
L2,Bol		5.4${}_{-5.2}^{+6.2}$ × 1029 erg s−1
RWD		0.0094 R☉

Notes. Numbers in brackets throughout the paper are 1σ uncertainties referring to the last significant digits quoted.

^a See Hachisu et al. (2004).

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The parameters of the system from the light-curve fitting are in an agreement with the estimates presented in Section 6. The inclination angle is i = 79.0°(2) according to the model. The effective temperature of the continuum-forming region is only 1200–1800 K. We estimate the mass transfer rate was 2.7 × 10⁻¹² M_⊙ yr⁻¹ during 2013 January observations. We confirm that the origin of double humps in the light curve is related to the spiral pattern in the accretion disk. However, its shape is affected slightly by the standard hot spot and emission from the opposite side spot. The last has the same effective temperature as the outer edge of the circular accretion disk. Thus it is not an additional hot spot in the classical understanding of this term. This extra area corresponds to a noncircular shape, and non-Keplerian velocities in the disk extended until the tidal limitation radius.

Other light curves of the object (like the one in the bottom panel of Figure 3) also can be reproduced by this model (the blue solid line in the bottom panel of Figure 3) by lowering the mass transfer rate ≈5 × 10⁻¹³ M_⊙ yr⁻¹, and some small adjustments of the parameters of the spiral pattern and spots.

8.1. Accretion Disk Spectrum

Using WD models with the determined temperatures, we extracted the spectra of the accretion disk by subtracting the underlying WD contribution to the observed spectra of EZ Lyn. This allowed us to recover even higher-order Balmer emission lines, which are almost not visible in the original spectra, being hidden inside of the WD absorption troughs. In Table 2 we have outlined different parameters of the continuum and of prominent Balmer lines that were measured from the accretion disk spectra. In all spectra, the disk contribution to the total system light is found to be very low, ranging between a few and 14% of the total system light in the Johnson V band, which is almost unaffected by strong emission lines. Due to a low level of the disk continuum, equivalent widths of the Balmer lines reach hundreds and even thousands angstroms. The examples of the disk spectra obtained before the first superoutburst (sdss-53090) and about 8 yr after the last superoutburst (not-58423) are shown in the bottom panel of Figure 5.

As can be seen from Table 2 and Figure 1 (bottom panel), both the line and continuum fluxes exhibited notable variations over the time. However, we did not detect correlation between any measured parameters. For example, all disk spectra obtained during the proper quiescent state (sdss-53090 and not-58423) have a similar continuum level, but the line fluxes in these spectra a factor of 2 different. In contrast, the line fluxes are similar in sdss-53090 and sdss-55590, and in sdss-55517 and not-58423, but the continuum in these pairs of spectra are very different.

The knowledge of stellar components in combination with the distance to EZ Lyn (the distance modulus m − M = 5.77^m(3)) allows the measurement of the absolute magnitude M_V,disk of the disk. Although M_V,disk varies greatly from epoch to epoch (Table 2), it consistently has a lower brightness than estimates of the disk's absolute magnitude in period-bounce candidates presented in Table 3 of Patterson (2011).

We can establish a conservative upper limit to the bolometric luminosity of the accretion disk to be L_d ≲ 1.5 × 10³⁰ erg s⁻¹ using the UV and NIR photometry obtained close in time with the NOT (JD 2458423) observations (V = 17.95) (Figure 8). We take into account the contribution from the donor star, which is very low though. Considering also the optical depth uncertainties in the disk, the upper limit of the bolometric luminosity is ${L}_{{\rm{d}}}/\cos (i)\,\lesssim 7.9\times {10}^{30}$ erg s⁻¹, where i = 79.0°. Thus, the lower limit on the luminosity corresponds to the fully transparent accretion disk in the continuum, while the upper one equates to the optical depth being close or reaching τ ∼ 2/3 at some point nearly its midplane. Such a luminosity corresponds to the mass accretion rate ${\dot{M}}_{\mathrm{acc}}$ ≃ (1.7–9.2) × 10¹³ g s⁻¹ = (0.29–1.5) × 10⁻¹² M_☉ yr⁻¹, from

$$\begin{eqnarray}&&{\dot{M}}_{\mathrm{acc}}=\displaystyle \frac{2{L}_{{\rm{d}}}{R}_{\mathrm{WD}}}{{{GM}}_{\mathrm{WD}}},\end{eqnarray} \tag{ 7 }$$

where the WD mass and radius were taken from Table 4.

Despite notable variations of the disk luminosity and the line fluxes over time, the Balmer decrement remains quite stable and relatively flat. On average, the Balmer decrement of the accretion disk in EZ Lyn is found to be Hα:Hβ:Hγ:Hδ = 1.61:1.0:0.76:0.59, with the standard deviation of each ratio being ≲0.15. Such Balmer decrement is comparable to values found in bright accretion disks of long period CVs (Williams 1983; Wagner et al. 1998; Neustroev et al. 2016). The flat decrement suggests that the emission lines are collisionally excited, rather than produced by photoionization, e.g., by the WD. Based on the non-LTE calculations of Drake & Ulrich (1980) and Williams (1991), it can be demonstrated that such flat decrements can be produced in the optically thin regions of the disk with the average density and temperature of logN₀ = 12.5(2) and T = (10–15) × 10³ K, respectively (Figure 11).

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Summarizing, we note that the disk luminosity L_d appears to be about 10 times larger than the X-ray luminosity, the latter is almost certainly produced by the boundary layer. This is consistent with the disk instability model according to which the mass accretion rate in the quiescent disk is expected to decrease steeply with decreasing radius (Cannizzo 1993; Ludwig et al. 1994). Hence, almost no matter is expected to reach the inner disk and the X-ray flux should indeed be very low.

8.2. Accretion Disk Parameters from Modeling of Emission-line Profiles

The emission lines from different data sets exhibit not only different intensity, but also noticeably different profiles. The double-peaked profile of emission lines is a result of the Doppler shift of emission due to Keplerian rotation of the accretion disk (Horne & Marsh 1986). It is well known that the separation between peaks in the double-peaked profiles is defined by the velocity of the outer rim of the disk V_out, which, in turn, depends on its radius r_out. The extent of the wings is determined by the ratio of the inner to the outer radii of the disk, R_in/R_out, whereas the shape of the line wings is controlled by the radial emissivity profile, which is commonly assumed to follow a power-law function of the form f(r) ∝ r^−b , where r is the radial distance from the accretor. These three accretion disk parameters—υ_out, b, and R_in/R_out—basically affect different parts of the line profile, and therefore they can be determined unambiguously (Smak 1981; Horne & Marsh 1986; Borisov & Neustroev 1997).

In order to derive the disk parameters in EZ Lyn, we fitted the averaged Hα profiles in all data sets using the method of Horne & Marsh (1986), which takes into account the Keplerian velocity gradient across the finite thickness of the disk. Examples of the application of this technique to the real data are given in Johnston et al. (1989), Orosz et al. (1994), Neustroev et al. (2002, 2014, 2016). We note that although the broad absorption line of Hα from the WD is not very deep, it may still affect the determination of the disk parameters (first of all, R_in/R_out). For this reason, for the modeling we used WD subtracted spectra. Figure 12 shows the averaged profiles of the Hα emission lines together with the corresponding model fits. The best-fitting model parameters are listed in Table 2. The errors were estimated with a Monte Carlo approach described in Borisov & Neustroev (1997). In all cases, they were within 5 km s⁻¹ for υ_out, 0.06 for b, and 0.02 for R_in/R_out. The results of the profile fits are very consistent and can be summarized as the following.

• 1.  
  The measured υ_out in all cases is close to the orbit-averaged value of ${\overline{\upsilon }}_{\mathrm{out}}=675$ km s⁻¹ of the tidally truncated disk, and always smaller than the velocity at the 3:1 resonance radius (υ_3:1 = 701 km s⁻¹ for i = 79°). Thus, the radius of the disk in EZ Lyn during all epochs of observations was always close to the tidal truncation radius, and definitely larger than the 3:1 resonance radius.
• 2.  
  Our model fits give quite consistent but relatively small values of the power-law index $\bar{b}=0.8(3)$ (Table 2), whereas this parameter in CVs with brighter accretion disks is usually found to be in the range of 1–2, rarely being less than 1.5 (Horne & Saar 1991). Thus, the radial emissivity profile in the weak disk of EZ Lyn is flatter.
• 3.  
  The ratio R_in/R_out is also consistent in all but one spectrum (sdss-53090). Excluding the latter, the mean value of R_in/R_out is 0.047(15). Coupling it with the mean value of υ_out = 663(23) km s⁻¹, we find the inner radius of the disk to be 1.16 × 10⁹ cm ≈ 2 R_WD = 0.017 R_⊙. Thus, R_in as seen in the Hα emission line appears much smaller than in the continuum light (R_in,cont ≈ 0.2R_⊙, see Table 4). Although the fit to the spectrum sdss-53090, obtained in quiescence before the first superoutburst, produces significantly larger R_in = 3.26 × 10⁹ cm ≈ 5 R_WD = 0.047R_⊙, it is still less than in the continuum.

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8.3. Doppler Tomography

Figure 13 presents the Hα Doppler maps of EZ Lyn produced using spectra taken in two different epochs. The 2008 observations (spm-54503 and spm-54537) were obtained 2 yr after the first superoutburst, when the object's brightness was V ∼ 17.5 or about 40% higher than in 2018 at V ≈ 17.9. The 2018 tomogram is based on not-58423 spectra. The maps were generated in the standard and the inside-out projections using codes developed by Spruit (1998) and Kotze et al. (2016), respectively. The Roche lobe of the secondary, the primary, and center mass positions, and the trajectory of the stream are calculated using the system parameters obtained in previous sections. The solid circle on the plot corresponds to the tidal limitation radius of the disk. The dotted-line circles mark the velocities with 500 km s⁻¹ steps. We defined the phase zero-point arbitrarily to place the hot spot position on the calculated stream trajectory for lack of pronounced WD eclipses or radial velocities of either of the stellar components.

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The appearance of the Doppler maps in both epochs is quite similar. ¹² There is a bright hot spot formed at the impact of the stream and the edge of the accretion disk. The maps show a nonuniform ring of disk emission extending in both data sets to the tidal limitation radius. Both maps displays an excess (of emitting elements of corresponding velocity) at the far side of the disk, opposite the donor star. This opposite-side region has a similar structure and spread. In particular, both maps show a relatively bright speck at υ_x ≈ − 600 km s⁻¹, υ_y ≈ − 600 km s⁻¹, and an extended ragged tail counterclockwise spreading slightly more than 90°. Interestingly, a similar second spot starting at the position about 120° contraclockwise from the standard hot spot was also observed in some Doppler maps of AM CVn systems: SDSS J124058.03-015919.2, SDSS J120841.96+355025.2, GP Com, and V396 Hya, Gaia14aae (Roelofs et al. 2005; Kupfer et al. 2013, 2016; Green et al. 2019b), which have a low mass ratio <0.08 and where the 2:1 resonance should be present. Green et al. (2019b) gives a short review of the origin of the second spot in the Doppler maps of AM CVns. We want to note that such a second spot can be seen not only in EZ Lyn and several AV CVns but also in another period bouncer V406 Vir (Aviles et al. 2010; Pala et al. 2019).

Indisputable similarity of the Doppler maps from two epochs despite different brightness validates Szkody et al. (2013b)'s postulation that the system brightness in the quiescence is a function of the WD heating, rather than a transformation in the disk's conditions. However, the source of the double humps gets stronger, when the system is brighter.

Assuming Keplerian velocities of particles in the disk, we transformed the Doppler (velocity) map into the XY plane (Figure 14, right panel) of the system and overlaid the model obtained from the light-curve fitting (Figure 14, left panel) on it (Figure 14, middle panel). The model is shown by color in the effective temperature scale (T) inside the range from 1000 K (black) to 4000 K (red), the intensity (I) converted from the Doppler map (not-58423) corresponds to the color bar white-red and linearly scaled from zero to one at a maximum. The color map in the XY plan clearly shows emission from the hot spot and clumpy structure of the disk with variable intensity along the disk position angle. There is an extended and relatively bright region at the side opposite to the hot spot. Although we do not expect that the radiation from the emission forming region must follow completely to spiral pattern responsible for the double-humped light curve observed in the continuum, nevertheless, we can see a qualitative agreement between them. The model spiral pattern escorts the distribution of brightness in the XY plane obtained from the Hα map. The relatively bright opposite-side emission region (to the standard hot spot position in the XY plane) is located at crosses the end of one spiral and the beginning of another spiral, and close to the second spot in the model. More probably, the Hα emission in this region is caused by spirals and non-Keplerian motion is expected here.

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Fraction of evolved CVs near the period minimum reaches 83% in line with population synthesis calculations (Pala et al. 2020, and references therein). This fact is now firmly established for our close vicinity (150 pc) thanks to the combination of SDSS survey and precise distances determined by the Gaia mission. Some of these short-period systems are period bouncers, distinguishing which is quite a difficult task. One crucial parameter that strongly differentiates systems before the period minimum and after is the mass accretion rate (Goliasch & Nelson 2015), which introduces subtle effect on the system's observational characteristics. Table 5 contains a list of well-established period bouncers. Some of them were proposed based on determination of their system parameters (Patterson 2011; Zharikov et al. 2013), others have been claimed stemming from analyses of light curves during superoutbursts (see Table 1 from Kimura et al. 2018). The general characteristics of those systems are a low optical luminosity at orbital periods longer than the period minimum (Paczynski & Sienkiewicz 1981), a relatively massive (M_WD ∼ 0.7–0.9 M_⊙) and a cool (10–16 kK) WD as the primary, and a brown dwarf as the secondary. Also, the mass ratios for all proposed candidates are close or below ≲0.08 and tend to decrease at larger orbital periods. One of the photometric features associated with period bouncers is presence of two humps per orbital period during the quiescence.

In this study we dissect the accretion disk of EZ Lyn in quiescence, confirming previous findings and providing new insights, which we summarize below:

• (1)  
  The Balmer decrement (as in Figure 5) is similar to bright accretion disks in longer period CVs with a higher accretion mass transfer. The emission lines are formed by a plasma of $\mathrm{log}{N}_{0}=12.5(2)$ density and T = (10–15) kK temperature, on average. The size of the accretion disk, where emission lines are formed, expands to the tidal truncation limit R_out ≈ 0.35 R_⊙ (Figure 15). It is in accordance with Neustroev & Zharikov (2020)'s conclusion that the the disk radius in short-period CVs is close or equal to the tidal truncation limit, and does not change whether in quiescence or outburst. The inner part of the disk extends inward down to R_in ≈ 5 R_WD ≈ 0.05 R_⊙. The high-velocity wings of emission lines are formed here.
• (2)  
  The optical continuum is formed predominantly by the radiation from the WD. Contribution of the accretion disk originates from the outer part (r ∈ [0.2 R_⊙, 0.35 R_⊙]) (Figures 8, 15) and is small (see Table 2) in the optical domain, however, it is significant in NIR sometimes. The effective temperature of the disk's continuum radiation is less than ≲2000 K. Kuulkers et al. (2011), Pala et al. (2019), among others, mention a cavity or evaporated fraction of the inner disk. This phenomenon can be interpreted also as a full transparency of this part in the continuum, contributing to the emission lines mainly. The outer part of the disk is divided into two emission layers. One is a hot region similar to the inner part of accretion disk where emission lines are forming. Another is an internal disk part where the main flux of the continuum originates. We show a schematic representation of this division in the Figure 15. The boundary between the fully optically thin inner part (r < 0.2 R_⊙) of the accretion disk and the continuum producing outer region is likely not fixed. Most probably the boundary is result of a transition from the LTE to non-LTE regime in the optically thin accretion disk (Dumont et al. 1991). The α parameter ¹³ of the disk in this case is 0.02 according to the Dumont et al. (1991, Equation (25)) with the rest of system parameters as in Table 4. The hot spot practically does not contribute to the continuum.The origin of the continuum-forming region, nevertheless remains debatable. Certainly, the hot, optically thin portion of the disk described as a hydrogen gas slab produces continuum. On the other hand, at the outer part of the disk the midplane zone can be cool and participate in the continuum formation. Dumont et al. (1991) notes that the temperature of the gas in optically thin, non-LTE regime with α less than 0.1 can drop below 5000 K. Meanwhile, Cannizzo & Wheeler (1984) found that, for α only moderately less than unity, there is only steady-state solution with T ≲ 2500 K at the outer edge of a standard dwarf novae disk. If the cooler portions have lower α, as in the case of systems around the period minimum, this tendency to form a distinct cool ring is exacerbated. The outermost rim of the disk can be optically thick and cool similar to WZ Sge as follows from the results of Howell et al. (2008) because at very low temperatures, T ≲ 2000 K, a dust absorption causes the opacity to increase. The atmosphere of the L-type brown dwarf secondary (T_eff ≤ 2400 K, Nakajima et al. 2004), which transfers a matter to the disk, is complex, and it is filled by molecules and dust grains with different chemical compositions (Helling & Woitke 2006; Helling et al. 2008; Allard et al. 2011). In any case, the double-humped orbital modulation in the light curve in the V band indicates the presence of a photosphere (an optical depth of τ ∼ 2/3) in the outer part of the accretion disk.
• (3)  
  Aviles et al. (2010) and subsequently Zharikov et al. (2013) argued that the double-humped light curves commonly observed in several period-bounce candidates are related to the spiral-like pattern in the outer rim of the disk formed as a result of 2:1 resonance (Lin & Papaloizou 1979; Osaki & Meyer 2002). The newly refined system parameters of EZ Lyn are in compliance with q ≲ 0.08, which is the crucial condition of 2:1 resonance. However, the mechanism of brightness changes is open to discussion. While the papers mentioned above consider the spirals themselves as sources of variability, Pala et al. (2019) demonstrate that the WD of V406 Vir has two hot spots on its surface, produced by the targeted accretion along the spiral arms, while the disk is truncated/evaporated close to the WD. V406 Vir and EZ Lyn were thought to be twins, but the former shows double humps in the light curve both in the optical and UV domains. The latter shows no such variability in the UV and hence hot spots on the WD are excluded. The double humps have to be formed in the disk. Our light-curve model combined with the Doppler tomography favor the idea that spirals in the accretion disk of EZ Lyn are responsible for modulation of the light curve. Contribution of the hot spot in the light curve is negligible, hence it cannot be a source of observed variability. In fact, there are three well-studied systems (V406 Vir, EZ Lyn, and V455 And) listed in Table 5 that show striking similarities in their spectral and photometric appearance, yet different underlying processes. In EZ Lyn we do not see truncated disk or influence of spiral arms on the WD, like in V406 Vir on the one hand. On the other hand, V455 And, also a bounce-back candidate with a double-humped light curve, is firmly established as an intermediate polar; its disk is definitely truncated by the magnetic field at the Alfvén radius. Its WD shows double spots, but modulating the UV flux with the double spin period frequency (Szkody et al. 2013a), contrary to what is observed in V406 Vir. The latter is definitely not a polar, hence the WD rotation should not be synchronized with the orbital period, and consequently the hot spots on the WD have different nature.
• (4)  
  We redetermined the system parameters and estimated that the accretion rate was $\dot{M}\approx$ 2.7 × 10⁻¹² M_⊙ yr⁻¹ in 2013. At the end of 2018, the accretion disk became completely optically thin. The disk contribution in continuum is negligible and $\dot{M}\lesssim 1.5\times$ 10⁻¹² M_⊙ yr⁻¹. At the end of 2020 and later, when double-humped light curve and the long-term variation of brightness came back, probably the accretion rate began to increase with new formation of the continuum-forming region.

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We are grateful to the anonymous referee for the valuable comments that helped improve this paper. Partly 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 in part with ALFOSC, which is provided by the Instituto de Astrofisica de Andalucia (IAA) under a joint agreement with the University of Copenhagen and NOTSA. This work is based upon observations carried out at the OAN SPM, Baja California, México. We thank the daytime and night support staff at the OAN SPM for facilitating and helping obtain our observations. This research has been was funded in a part by the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan (grant No. AP08856419). Crimean observations in 2019–2021 are supported by the grant of SRF (project number 19-72-10063). S.Z. acknowledge PAPIIT-DGAPA-UNAM (grant IN102120). K.L.P. acknowledges funding from the UK Space Agency. S.K. acknowledges Al-Farabi Kazakh National University for the postdoc position. We thank Dr. V. Neustroev for help with observations, data analysis, and useful discussions.