Constraining the evolution of cataclysmic variables via the masses and accretion rates of their underlying white dwarfs

ABSTRACT

1 INTRODUCTION

Cataclysmic variables (CVs) are compact interacting binaries in which a white dwarf is accreting mass from a low-mass star via Roche lobe overflow (e.g. Warner 1995). CVs descend from main-sequence binaries in which the more massive star (the primary) evolves first and leaves the main sequence. Following its expansion, the primary star fills its Roche lobe and starts unstable mass transfer on its less massive companion (the secondary), leading to the formation of a common envelope, i.e. a shared photosphere engulfing both stars (e.g. Paczynski 1976; Ivanova et al. 2013). The two stars transfer orbital energy to the envelope, which is rapidly expelled, leaving behind a post-common envelope binary composed of the core of the giant primary, which evolves into a white dwarf, and a low-mass secondary star. Owing to subsequent orbital angular momentum losses, mainly via magnetic braking (which arises from a stellar wind associated with the magnetic activity of the secondary, e.g. Mestel 1968; Verbunt & Zwaan 1981) and gravitational wave radiation (e.g. Paczyński 1967), the post-common envelope binary evolves into a semi-detached configuration, becoming a CV.

During the CV phase, the white dwarf response to the mass accretion process is the subject of a long-standing debate. Many binary population studies predict an average mass of CV white dwarfs of |$\langle M_\mathrm{WD} \rangle \simeq 0.5\, \mathrm{M_\odot }$| (e.g. de Kool 1992; Politano 1996), which is lower than that of single white dwarfs (⁠|$\langle M_\mathrm{WD} \rangle \simeq 0.6\, \mathrm{M_\odot }$|⁠, Koester, Schulz & Weidemann 1979; Liebert et al. 2005; Kepler et al. 2007). This is because it is expected that the core growth of the primary is halted by the onset of the common envelope phase. Moreover, once a CV is formed, the material piled up at the white dwarf surface is compressed by the strong gravitational field of the star, leading periodically (typically on time-scale of ten/hundred thousands of years) to the occurrence of classical nova eruptions. These are the result of the thermonuclear ignition of the accreted material on the white dwarf surface and, during these explosions, different theoretical models predict that the accreted material (e.g. Yaron et al. 2005) and part of the underlying core of the white dwarf (Gehrz et al. 1998; Epelstain et al. 2007) should be ejected in the surrounding space, thus preventing the white dwarf from growing in mass. However, white dwarfs in CVs have been found to be significantly more massive than binary population models predicted (e.g. de Kool 1992; Politano 1996), with early work showing the average mass of CV white dwarfs to lie in the range |$\langle M_\mathrm{WD} \rangle \simeq 0.8\!-\!1.2\, \mathrm{M}_\odot$| (Warner 1973; Ritter 1987). This result was originally interpreted as an observational bias because (i) the higher the mass of the white dwarf, the larger the accretion energy released per accreted unit mass, and (ii) for a fixed donor mass, more massive white dwarfs have larger Roche lobes that can accommodate larger, and hence brighter (especially at optically wavelengths) accretion discs. Therefore, CVs hosting massive white dwarfs were expected to be more easily discovered in magnitude limited samples (Ritter & Burkert 1986). Later on, Zorotovic, Schreiber & Gänsicke (2011) reviewed the masses of CV white dwarfs available in the literature, showing that the average mass of CV white dwarfs is |$\langle M_\mathrm{WD} \rangle = 0.83 \pm 0.23\, \mathrm{M}_\odot$|⁠. Using the large and homogeneous sample of systems (Szkody et al. 2011, and references therein) discovered by the Sloan Digital Sky Survey (SDSS; York et al. 2000), the authors also demonstrated that there is a clear trend in disfavouring the detection of massive white dwarfs (which are smaller and less luminous than low-mass white dwarfs for the same |$\rm{T_\mathrm{eff}}$|⁠) and therefore the high average mass of CV white dwarfs cannot be ascribed to an observational bias.

Following this result, several authors investigated the possibility that CV white dwarfs could efficiently retain the accreted material and grow in mass, either by ejecting less material than they accrete, via quasi-steady helium burning after several nova cycles (Hillman et al. 2016), or through a phase of thermal time-scale mass transfer during the pre-CV phase (Schenker et al. 2002). Wijnen, Zorotovic & Schreiber (2015) argued that mass growth during nova cycles cannot reproduce the observed distribution. In addition, these authors found that, even though the white dwarf mass could increase during thermal time-scale mass transfer, the resulting CV population would harbour a much higher fraction of nuclear evolved donor stars than observed, and thus they concluded that mass growth does not seem to be the reason behind the high masses of CV white dwarfs. A general consensus on the ability of the white dwarf to retain the accreted mass has not been reached yet (Hillman et al. 2020; Starrfield et al. 2020). Moreover, neither mechanism is able to explain the observed CV white dwarf mass distribution without creating conflicts with other observational constraints.

A promising alternative solution proposed throughout the last couple of years assumes that the standard CV evolution model is incomplete. Schreiber, Zorotovic & Wijnen (2016) suggested that consequential angular momentum loss (CAML) is the key missing ingredient. This sort of angular momentum loss arises from the mass transfer process itself and acts in addition to magnetic braking and gravitational wave radiation. Schreiber et al. (2016) developed an empirical model (eCAML) in which the strength of CAML is inversely proportional to the white dwarf mass, leading to dynamically unstable mass transfer in most CVs hosting low-mass white dwarfs (⁠|$M_\mathrm{WD} \lesssim 0.6\, \mathrm{M_\odot }$|⁠). The majority of these systems would not survive as semi-detached binaries but the two stellar components would instead merge into a single object (see also Nelemans et al. 2016). The main strength of the eCAML model is that it can also solve other disagreements between theory and observations, such as the observed CV space density and orbital period distribution (Zorotovic & Schreiber 2017; Belloni et al. 2018, 2020; Pala et al. 2020), without the requirement of additional fine tuning. However, the exact physical mechanism behind this additional source of angular momentum loss and the reason for its dependence on the white dwarf mass are still unclear.

Finally, it has to be considered that, while observational biases can be ruled out, it is more difficult to assess the presence of systematics affecting Zorotovic et al. (2011)’s results, which were based on a sample of only 32 systems with accurate mass measurements, 22 of which were derived from the analysis of their eclipse light curves. Given that the previous results were mainly based on one methodology, it is necessary to increase the number of systems with an accurate mass measurement and diversify the methods employed in order to confirm the inferred high masses of CV white dwarfs, which remains one of the biggest and unresolved issues for the theoretical modelling of CV evolution.

An alternative method to measure CV white dwarf masses consists of the analysis of their ultraviolet spectra. The ultraviolet waveband is optimal for studying the underlying white dwarfs as they are relatively hot (⁠|$\rm{T_\mathrm{eff}} \gtrsim 10\, 000\,$| K, Sion 1999; Pala et al. 2017) and dominate the emission at these wavelengths, while the optical waveband is contaminated by the emission from the accretion flow and the companion star. From the knowledge of the distance to the system, the white dwarf radius (R_WD) can be derived from the scaling factor between the best-fitting model and the ultraviolet spectrum. Under the assumption of a mass–radius relationship, it is then possible to measure the white dwarf mass. Another possibility is to perform a dynamical study. The radial velocities of the white dwarf and the donor allow to infer the white dwarf gravitational redshift that, combined with a mass-radius relationship, provides the mass of the white dwarf.

Over the last 30 years, the Hubble Space Telescope (HST) has proven to be essential for the study of CVs, delivering ultraviolet observations¹ for 193 systems. Nonetheless, in the past, only a handful of objects had mass measurements derived from the analysis of their ultraviolet data because of the lack of accurate CV distances. In this respect, the Gaia space mission represents a milestone for CV research, delivering accurate parallaxes for a large number of these interacting binaries, finally enabling a quantitative analysis of the ultraviolet data obtained over the past decades.

We here analyse the HST observations of 43 CVs for which the white dwarf is the dominant source of emission in the ultraviolet wavelength range, and for which Gaia parallaxes from the Third Early Data release (EDR3) are available. To provide an additional independent determination to test our results and assess the presence of systematics, we complement this data set with optical phase-resolved observations obtained with X-shooter mounted on the Very Large Telescope (VLT), which provides independent dynamical mass measurements. We present this large CV sample, which doubles the number of objects with accurate M_WD measurements, providing new constraints on the response of the white dwarf to the mass accretion process, and for the further development of models for CV evolution.

2 SAMPLE SELECTION AND OBSERVATIONS

2.1 Ultraviolet observations

Among the 193 systems in the HST ultraviolet archive, 191 have a Gaia EDR3 parallax (Table 45 of the online material) and have been observed with either the Goddard High Resolution Spectrograph (GHRS), the Faint Object Spectrograph (FOS), the Space Telescope Imaging Spectrograph (STIS), or the Cosmic Origin Spectrograph (COS) with a setup suitable for our analysis, i.e. (at least) the full coverage of the Ly α (⁠|$1100\!-\!1600\,$| Å) from the white dwarf photosphere with a resolution of R ≃ 1000–3000. Together with a signal-to-noise ratio (SNR) of at least ≃10 and the knowledge of the distance to the systems, this setup allows an accurate determination of the white dwarf effective temperatures, chemical abundances, and masses (Gänsicke et al. 2005).

CVs are characterized by an intrinsic variable behaviour that can affect the analysis of their ultraviolet data. A large fraction of CVs spend most of their time in a quiescent state, in which the accretion rate on to the white dwarf is very low, though the mass buffered in the accretion disc is slowly built up. During this phase, the white dwarf dominates the spectral appearance of the system and, at ultraviolet wavelengths, is recognisable from broad Ly α absorption centred on |$1216\,$| Å. The profile of Ly α changes with T_eff, becoming more defined and narrower in the hotter white dwarfs, while the continuum slope of the spectrum becomes steeper (Fig. 1). Periods of quiescence are interrupted by sudden brightenings with amplitudes of 2–5 mag and, occasionally up to 9 mag (Maza & Gonzalez 1983; Warner 1995; Templeton 2007), called dwarf nova outbursts, when a fraction of the disc mass is rapidly drained on to the white dwarf. These outbursts arise from thermal-viscous instabilities in their accretion discs, causing a variation in the mass transfer rate through them (Osaki 1974; Meyer & Meyer-Hofmeister 1984; Hameury et al. 1998). Immediately after the occurrence of an outburst, the disc is hot and while it cools down it can partially or totally outshine the emission of the white dwarf even in the ultraviolet (top panel of Fig. 2).

Figure 1.

Sample HST spectra showing, from top to bottom, four typical quiescent CVs (SDSS J075507.70+143547.6, CY UMa, U Gem, and RX And) observed with different HST instruments, and one eclipsing system (IY UMa) whose spectrum is characterized by the presence of the iron-curtain. The geocoronal emission lines of Ly α (⁠|$1216\,$| Å) and O i (⁠|$1302\,$| Å, not always detected) are plotted in grey.

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Figure 2.

Sample HST spectra of a dwarf nova observed four days after a disc outburst (PU CMa, top panel) and a nova-like star observed in its normal high state (OR And, bottom panel). The spectra are dominated by the disc emission rather than the white dwarf. The geocoronal emission lines of Ly α (⁠|$1216\,$| Å) and O i (⁠|$1302\,$| Å) are plotted in grey.

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Following an outburst, the white dwarf is heated as a consequence of the increased infall of material, and this can possibly give rise to a non-homogeneous distribution of the temperature across its visible surface. Heated accretion belts or hot spots can dominate the overall ultraviolet emission of the system and the resulting temperature gradient results in the white dwarf radius being underestimated (e.g Toloza et al. 2016; Pala et al. 2019). The presence of a hot spot can be easily unveiled thanks to the modulation it introduces in the light curve of the system. In contrast, an equatorial accretion belt (Kippenhahn & Thomas 1978) can be difficult to detect since, being symmetric with respect to the rotation axis of the white dwarf, it does not cause variability on the white dwarf spin period (e.g GW Lib; Toloza et al. 2016). Therefore, the analysis of the ultraviolet data obtained after a disc outburst provides only an upper limit on the white dwarf effective temperature and radius and, in turn, only a lower limit on its mass.

Additionally, members of a sub-class of CVs, known as nova-like systems, are characterized by high mean mass-transfer rates which usually keep the discs in a stable hot state, equivalent to a dwarf nova in permanent outburst. In this high state, the disc dominates the spectral appearance even in the far-ultraviolet, preventing a direct detection of the white dwarf (bottom panel of Fig. 2). However, occasionally, it is thought that as a consequence of starspots appearing in, or migrating into the tip of the donor star at the first Lagrangian point (Livio & Pringle 1994), the mass transfer rate drops (low state) and unveils the white dwarf (e.g. Gänsicke & Koester 1999; Knigge et al. 2000; Hoard et al. 2004). These low states typically last for days up to years (e.g Rodríguez-Gil et al. 2007) and provide a window in which the white dwarf parameters can be measured (e.g. Gänsicke et al. 1999; Knigge et al. 2000; Hoard et al. 2004).

While some nova-likes can be magnetic, similar behaviour is observed in highly magnetic CVs. A significant number (28 per cent, Pala et al. 2020) of CVs contain strongly (⁠|$B \gtrsim 10\, \mathrm{MG}$|⁠) magnetic white dwarfs, whose magnetism suppresses the formation of an accretion disc and forces the accretion flow to follow the field lines. The strong field of the white dwarf may have a deep impact on the evolution of the system (Schreiber et al. 2021). Similar to nova-likes, magnetic CVs are also characterized by alternating between high and low states, on time-scales of months to years. During high states, mass accretion is stable and the ultraviolet emission is dominated by a hot polar cap close to one (or both) the magnetic pole(s) of the white dwarf. During low states, the hot-caps emission is greatly weakened and the white dwarf dominates the spectral appearance of the system. Possibly, low states in magnetic CVs are also triggered by a donor stellar spot passing around the first Lagrangian point region (Hessman, Gänsicke & Mattei 2000). In low states, time-resolved observations are required in order to account for the possible contribution from the accretion cap and to obtain accurate white dwarf parameters.

Finally, CVs experience (at least once in their life) powerful classical nova eruptions due to thermonuclear runaways at their surface. CVs that have been observed to undergo these eruptions are known as novae and their ultraviolet spectra are dominated by the emission from a hot highly variable component (e.g. Cassatella, Altamore & González-Riestra 2005), whose origin is still not clear, that prevents the direct detection of the white dwarf.

To account for the high variable nature of CVs, we inspected the archival ultraviolet HST data and discarded (i) dwarf novae observed during or immediately after a disc outburst; (ii) nova-likes in high state and novae and (iii) polars lacking time-resolved observations with sufficient orbital phase coverage, necessary to determine a spectrum of the underlying white dwarf (see e.g. Gänsicke et al. 2006). In addition, we also discarded systems hosting a nuclear evolved donor. These CVs descend from binaries that underwent a thermal-time-scale mass transfer phase (Schenker et al. 2002) and can be easily identified from their enhanced N v/C iv line flux ratios (Gänsicke et al. 2003). In these systems, the white dwarf accretes helium rich material from its companion and this anomalously large helium abundance can cause an asymmetrical broadening of the blue wing of the Ly α (Gänsicke et al. 2018), thus affecting the estimates of the white dwarf surface gravity and temperature (Toloza et al., in preparation).

The final sample consists of 43 systems² and a log of their spectroscopic observations is presented in Table 1, while the different observational setups are listed in Table 2. Finally, the full list of CVs observed with HST is provided in Section 5 of the online material.

2.1.1 Quality of the Gaia data

Gaia astrometric solutions, and thereby parallaxes, are known to be affected by systematics arising from imperfections in the instrument and data processing methods (Lindegren et al. 2020). The mean value of the systematic error, the so-called parallax zero-point ϖ_zp, can be modelled according to the ecliptic latitude, magnitude, and colour of each Gaia EDR3 source. We employed the python code provided by the Gaia consortium³ to compute ϖ_zp for our targets, and corrected their parallaxes by subtracting the estimated zero-point to the quoted Gaia EDR3 parallaxes. This correction ranges from 0.5 |$\mu$|as, in the case of V1108 Her, to −43 |$\mu$|as for U Gem.

Together with the main kinematic parameters (positions, proper motions, and magnitudes), which are used to derive the astrometric solution for each source, Gaia EDR3 also provides a series of ancillary parameters that can be used to evaluate the accuracy of this solution. Among the most relevant ones is the astrometric_excess_noise, which represents the error associated with the astrometric modelling (see Lindegren et al. 2012) and that, ideally, should be zero. Following Pala et al. (2020), we verified that the sources in our sample have reliable parallaxes by satisfying the condition astrometric_excess_noise < 2.

Converting parallaxes into distances is not always trivial as the mere inversion of the parallax can introduce some biases in the distance estimate, especially when the fractional error on the parallax is larger than 20 per cent (see e.g. Bailer-Jones 2015; Luri et al. 2018), which is the case for two systems in our sample, AL Com and LL And. Their large uncertainties are most likely related to their intrinsic faintness (G = 19.7, G = 20.1, respectively) since the other Gaia parameters that flag possible issues with the astronometric solution (astrometric_excess_noise and RUWE, described below) are within the range expected for well-behaved sources. We therefore computed the distance to each CV using a statistical approach, in which we assumed an exponentially decreasing volume density prior and a scale height h following the method⁴ described in Pala et al. (2020). Typically, the distances computed with the two methods differ by less than one per cent, the only exception being LL And with a difference of five per cent (Fig. 3).

Figure 3.

Comparison between the distances to our targets computed as the inverse of the parallax (ϖ⁻¹) and using a statistical approach via the assumption of an exponentially decreasing volume density prior. The data are colour coded according to the parallax fractional error. The fractional difference, defined as (D − ϖ⁻¹)/ϖ⁻¹, between the two methods is typically less than one per cent (red dashed lines in the bottom panel).

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Another important parameter to assess the reliability of the parallaxes is the normalized unit weight error (RUWE). This represents the square root of the normalized chi-square of the astrometric fit, scaled according to the source magnitude G, its effective wavenumber |$\nu_\mathrm{eff}$| and its pseudocolour |$\hat{\nu }_\mathrm{eff}$| (see Lindegren et al. 2020 for more details). Ideally, for well-behaved sources,⁵RUWE < 1.4. However, we noticed that for the system in our sample with the largest RUWE, AM Her (RUWE = 2.8), the distance derived from its Gaia parallax (⁠|$88.1 \pm 0.4\,$| pc) is consistent with the distance estimated by Thorstensen (2003), |$79^{+9}_{-6}\,$|pc. Similarly, in the case of U Gem, another system with high weight error (RUWE = 2.4), its Gaia parallax (ϖ = 10.75 ± 0.03 mas) and corresponding distance (⁠|$93.4 \pm 0.5\,$| pc) are in good agreement, respectively, with the parallax measurements obtained using the HST Fine Guidance Sensors by Harrison et al. (2000) (ϖ = 10.30 ± 0.50 mas) and Harrison et al. (2004) (ϖ = 9.96 ± 0.37 mas), and with the distance estimated by Beuermann (2006), |$97\pm \, 7$|pc. These large values of RUWE are most likely related to colour variations of the systems during different Gaia observations, caused either by the occurrence of low and high states in AM Her (Gänsicke et al. 2006), and disc outbursts in U Gem. Nonetheless, since their Gaia astrometry still provides reliable distances, we decided to not apply any cut on this parameter for the remaining systems, which all have lower RUWE values.

2.2 Optical spectroscopy

We obtained complementary phase-resolved spectroscopy with X-shooter (Vernet et al. 2011) of the 22 targets from the HST sample that are visible from the Southern hemisphere and in which (i) the white dwarf dominates the of emission in the ultraviolet and (ii) for which no dynamical study has been carried out before.

X-shooter is an échelle spectrograph located at the Cassegrain focus of UT2 of the VLT at the European Southern Observatory (ESO) in Cerro Paranal (Chile). It is equipped with three arms: blue (UVB, |$\lambda \simeq 3000\!-\!5595\,$| Å), visual (VIS, |$\lambda \simeq 5595\!-\!10\, 240\,$| Å), and near-infrared (NIR, |$\lambda \simeq 10\, 240\!-\!24\, 800\,$| Å), with a medium spectral resolution (⁠|$R \simeq 5000\!-\!10\, 000$|⁠). For each arm, the slit width was chosen to best match the seeing and the exposure times were set with the aim to optimize the SNR and, at the same time, to minimize the orbital smearing. At the time of the observations the atmospheric dispersion correctors of X-shooter were broken and hence the slit angle was reset to the parallactic angle position after one hour of exposures. The data were reduced using the Reflex pipeline (Freudling et al. 2013). To account for the well-documented wavelength shift between the three arms,⁶ we used theoretical templates of sky emission lines to calculate the shift of each spectrum with respect to the expected position. We then applied this shift together with the barycentric radial velocity correction to the data. Finally, a telluric correction was performed using molecfit (Kausch et al. 2015; Smette et al. 2015).

In the spectra of three (AX For, IR Com, and V1108 Her) of the 22 CVs observed with X-shooter, we identified the Mg ii absorption line at 4481 Å that originates in the white dwarf photosphere, and several absorption features arising from the secondary photosphere, including Na i (⁠|$11\, 381/11\, 403$| Å), K i (⁠|$11\, 690/11\, 769$| and |$12\, 432/12\, 522$| Å). The K i and Mg ii lines, were used to track the motion of the two stellar components and to reconstruct their radial velocity curves from which the mass of the white dwarf can be determined. A log of the spectroscopic observations is presented in Table 3.

In the remaining 19 systems, we identified only signatures of either the white dwarf or the secondary, and in some cases of neither of them, and the analysis of these objects will be presented elsewhere.

3 METHODS

3.1 Light curve analysis

Throughout the duration of the individual HST observations (typically a few hours), CVs can show different types of variability, such as eclipses, modulations due to the white dwarf rotation, white dwarf pulsations, double humps, and brightenings (e.g. Szkody et al. 2002b; Araujo-Betancor et al. 2003; Toloza et al. 2016; Szkody et al. 2017; Pala et al. 2019). Eclipses are particularly important since they allow a white dwarf mass measurement based only on geometrical assumptions to be obtained. In contrast, pulsations and brightenings reflect the presence of hot spots and, more generally, of a gradient in temperature over the visible white dwarf surface. When in view, the emission of the hot spots can dominate the overall ultraviolet emission of the system making the white dwarf look hotter and affecting both the temperature and mass estimates (Toloza et al. 2016; Pala et al. 2019). Therefore, it is important to remove the contribution of these spurious sources in order to obtain an accurate mass measurement.

The HSTtime–tag data allow us to reconstruct a 2D image of the detector, where the dispersion direction runs along one axis and the spatial direction along the other, which can be used to reconstruct the light curve of the observed system. For each CV, we masked the geocoronal emission lines from Ly α (centred at 1216 Å) and O i (centred at 1300 Å) as well as all the most prominent emission features from the accretion disc, which are not representative of the white dwarf. Using five-second bins and following the method described in Pala et al. (2019), we then extracted the light curve of each target in counts per second.

For the objects that did not exhibit any significant variability during their HST observations, the data from all the orbits were summed to produce an average ultraviolet spectrum. We discuss, in what follows, the eclipsing systems while the remaining six CVs that showed some level of variability within the time-scale of the HST observations are discussed in Section 1 of the online material.

3.1.1 Eclipsing systems

To this end, one of the most direct method consists in measuring the radial velocity amplitudes (K_WD and K_donor) of the two stellar components from phase-resolved spectroscopic observations. These quantities provide the system mass ratio q = K_WD/K_donor = M_donor/M_WD, which allows constraining the size of the Roche lobe of the companion star (Eggleton 1983). Under the assumption that the donor is Roche lobe filling, the degeneracy in the three quantities can be lifted. In this case, a fit to the eclipse light curve provides R_WD/a which, combined with a mass radius relationship and Kepler’s third law, provides a measurement of the white dwarf mass (see e.g. Feline et al. 2005; Littlefair et al. 2006; Savoury et al. 2011; McAllister et al. 2019).

We detected the white dwarf eclipse in the COS light curves of four CVs, IR Com, IY UMa, SDSS J103533.02+055158.4, and SDSS J150722.30+523039.8. However, the data quality and orbital coverage allowed us to perform a fit to the eclipse light curve only in the cases of IR Com and SDSS J150722.30+523039.8. The remaining eclipsing systems are discussed in Section 1.0.1 of the online material.

We used the lcurve tool⁷ (see Copperwheat et al. 2010, for a detailed description of the code) to perform the light curve modelling and define the binary star model that best reproduces the observed eclipse. For IR Com we assumed the mass ratios derived from the radial velocity amplitudes from Section 3.3, q = 0.016 ± 0.001. For SDSS J150722.30+523039.8 we used the mass ratio from Savoury et al. (2011), q = 0.0647 ± 0.0018, which has been derived from the analysis of the optical light curve of the system. For both CVs, we assumed the white dwarf effective temperatures derived in Section 3.2, which are used by lcurve to estimate the flux contribution from the white dwarf. We kept the inclination i, R_WD/a and the time of middle eclipse T₀ as free parameters. The best-fitting models are shown in the insets in Fig. 4 and returned i = 80.5 ± 0.3 and R_WD/a = 0.00956± 0.0002 for IR Com and i = 83.5 ± 0.3 and R_WD/a = 0.0185 ± 0.0006 for SDSS J150722.30+523039.8, respectively. The R_WD/a ratios, combined with the white dwarf mass–radius relationship⁸ (Holberg & Bergeron 2006; Tremblay, Bergeron & Gianninas 2011) and Kepler’s third law, provide |$M_\mathrm{WD} = 0.989 \pm 0.003\, \mathrm{M_\odot }$| for IR Com and |$M_\mathrm{WD} = 0.83^{+0.19}_{-0.15}\, \mathrm{M_\odot }$| for SDSS J150722.30+523039.8 (Table 6).

Figure 4.

HST/COS light curves of IR Com (left) and SDSS J150722.30+523039.8 (right). The absence of any contamination from the bright spot allows us to fit the eclipse light curve and measure the mass of the white dwarf in these systems. The insets show a close-up of the eclipse of IR Com and of the average phase-folded eclipse light curve of SDSS J150722.30+523039.8, along with the best-fitting models (red).

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It is worth mentioning at this point that the eclipse of the white dwarf in DW UMa was detected in the STIS time–tag data. This light curve has already been analysed by Araujo-Betancor et al. (2003), which derived a white dwarf mass of |$M_\mathrm{WD} = 0.77 \pm 0.07\, \mathrm{M_\odot }$|⁠. Finally, DV UMa is also eclipsing but, since the data were acquired as snapshot, the light curve of the eclipse is not available.

3.2 Ultraviolet spectral fitting

To perform the spectral fit to the ultraviolet data, we generated a grid of white dwarf synthetic atmosphere models using tlusty and synspec (Hubeny 1988; Hubeny & Lanz 1995), covering the effective temperature range |$T_\mathrm{eff} = 9000 \!-\! 70\, 000\,$| K in steps of |$100\,$| K, and the surface gravity range log (g) = 6.4–9.5 in steps of 0.1 (where g is expressed in cgs units). As discussed by Pala et al. (2017), using a single metallicity is sufficient to account for the presence of the metal lines and possible deviations of single element abundances from the overall scaling with respect to the solar values do not affect the results of the fitting procedure. We therefore fixed the metal abundances to the values derived from the analysis of the same HST data by previous works (see Table 5 and references therein).

The HST ultraviolet spectra are contaminated by geocoronal emission of Ly α and O i (1302 Å). The first is always detected and we masked the corresponding wavelength range for our spectral analysis. In the case of O i, the related wavelength range was masked only when the emission was detected in the spectrum. Moreover, CV ultraviolet spectra show the presence of an additional continuum component, which contributes |$\simeq 10\!-\!30\,$| per cent of the observed flux. The origin of this additional emission source is unclear and it has been suggested that it could arise from either (i) the disc, or (ii) the hot spot where the ballistic stream intersects with the disc or (iii) the interface region between the disc and the white dwarf surface (e.g. Long et al. 1993; Godon et al. 2004; Gänsicke et al. 2005). In the literature, different approximations have been used to model the emission of this additional component, such as a blackbody, a power law or a constant flux (in F_λ). As discussed by Pala et al. (2017), these assumptions represent a very simplified model of the additional continuum contribution and it is likely that none of them provides a realistic physical description of this emission component. These authors also showed that, when only a limited wavelength coverage (⁠|$1105\!-\!1800\,$| Å) is available, it is not possible to statistically discriminate among the three of them, and they all result in fits of similar quality for the white dwarf. We therefore decided to use a blackbody, which is described by two free parameters (a temperature and a scaling factor), because, in the limited wavelength here considered, its tail approximates both the power law and the constant flux cases.

The only exceptions are the eclipsing systems (discussed below) and GD 552. For the latter, additional data obtained with the G230L grating, covering the wavelength range |$1650\!-\!3150\,$| Å, are available. As already noticed by Unda-Sanzana et al. (2008), the additional second component in the near-ultraviolet flux of GD 552 shows a clear dependence on wavelength (orange line in Fig. 5). The observed slope could arise from either an optically thin emission region, or an optically thick accretion disc, or a bright spot. All cases are expected to display a power-law distribution (which, for the optically thick disc and the bight spot, results from the approximation with a sum of blackbodies) and, given the wide wavelength coverage available for this object, a single blackbody would represent a poor approximation for the additional emission component. Therefore, in the case of GD 552, we assumed a power-law, which is described by two free parameters (a power-law index and a scaling factor).

Figure 5.

HST spectrum of GD 552 obtained from the combination of the STIS/G140L far-ultraviolet (blue) and STIS/G230L (black) data (the grey bands mask the geocoronal emission lines of Ly α at |$1216\,$| Å and of O i at |$1302\,$| Å, while the gaps at |$\simeq 1950\,$| Å and |$\simeq 2600\,$| Å correspond to data with bad quality flags). The best-fitting model (red) is composed of the sum of the white dwarf emission (grey), the emission lines approximated as Gaussian profiles (green), and an additional second component (orange) which shows a clear dependence on wavelength. This suggests that this additional emission component in the system is either an optically thin emission region, or an optically thick accretion disc, or a bright spot. These cases are expected to display a power-law distribution and, in our fitting procedure, we assumed a power law to model the emission of this additional component in GD 552.

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The analysis of the eclipsing systems is complicated by the presence of the so-called ‘iron curtain’, i.e. a layer of absorbing material extending above the disc which gives rise to strong absorption features (see e.g. the spectrum of IY UMa in Fig. 1), mainly a forest of blended Fe ii absorption lines (Horne et al. 1994). These veil the white dwarf emission, making it difficult to establish the actual flux level (necessary to constrain the white dwarf radius). In addition, these lines modify the overall slope of the spectrum as well as the shape of the core of the Ly α line, which are the tracers for the white dwarf T_eff. Out of the six eclipsing systems in our sample, two are strongly affected by the veiling gas: IY UMa and DV UMa. Following the consideration by Pala et al. (2017), in the spectral fitting of these CVs, we included two homogeneous slabs, one cold (⁠|$T_\mathrm{curtain} \simeq 10\, 000\,$| K) and one hot (⁠|$T_\mathrm{curtain} \simeq 80\, 000\,$| K). We generated two grids of monochromatic opacity of the slabs using synspec, one covering the effective temperature range |$T_\mathrm{eff} = 5000 \!-\! 25\, 000\,$| K and the other the range |$T_\mathrm{eff} = 25\, 000 \!-\! 120\, 000\,$| K, both in steps of 5000 K. Both grids covered the electron density range n_e = 10⁹–10²¹ cm⁻³ in steps of 10³ cm⁻³ and the turbulence velocity range |$V_\mathrm{t} = 0\!-\!500\, \mathrm{km\, s}^{-1}$| in steps of |$100\, \mathrm{km\, s}^{-1}$|⁠. These models, combined with the column densities (N_H, for which we assumed a flat prior in the range |$10^{17} \!-\! 10^{23}\, \mathrm{cm}^{-2}$|⁠), return the absorption due to the curtain. Given the large number of free parameters involved in the spectral fitting of the eclipsing systems, we chose to use a constant flux (in F_λ) to approximate the additional continuum component, as this is the simplest approximation and introduces only one more additional free parameter in the fitting procedure.

Finally, for all systems, whenever detected, we included the emission lines arising from the accretion disc as Gaussian profiles, allowing three free parameters: amplitude (f_em), wavelength (λ_em), and width (σ_em). As shown by Pala et al. (2017), by including or masking the disc lines has no influence on the result but their inclusion allowed us to use as much of the data as possible.

We performed the spectral fit using the Markov chain Monte Carlo (MCMC) implementation for python, emcee (Foreman-Mackey et al. 2013).

We scaled the models according to the distance to the system, computed as described in Section 2.1.1, and reddened them according to the extinction values reported in Table 1. We used the STructuring by Inversion the Local Interstellar Medium (Stilism) reddening map (Lallement et al. 2018) to derive the E(B − V). For those objects not included in the Stilism reddening maps, we used instead the 3D map of interstellar dust reddening based on Pan-STARRS 1 and 2MASS photometry (Green et al. 2019).

The free parameters of the fit and their allowed range of variations are listed in Table 4, and we assumed a flat prior in the ranges covered by the corresponding grid of models. We assumed the mass–radius relation of Holberg & Bergeron (2006), Tremblay et al. (2011) and constrained the parameters describing the blackbody (BB) and constant additional second components to be positive. In the case of the power-law (PL) additional second component, we only constrained its scaling factor to be positive.

The accuracy derived from the statistical uncertainties are typically two and four per cent for |$\rm{T_\mathrm{eff}}$| and R_WD, respectively. These uncertainties together provide an accuracy on the white dwarf masses of typically |$0.03\, \mathrm{M_\odot }$|⁠. However, the real uncertainties are dominated by systematic effects, which are discussed in the following Section.

3.2.1 Uncertainty estimate

As discussed by Pala et al. (2017), we can rule out the presence of systematics arising from instrument flux calibration issues as well as any noticeable contamination from additional Ly α absorption from the second emission component and/or interstellar gas along the line of sight. Moreover, the uncertainties related to the unknown nature of the second component are smaller than the statistical errors from the fitting procedure (Pala et al. 2017), therefore their effect is already accounted for in the error balance from the previous Section.

The remaining sources of uncertainties are hence those related to the precision on the Gaia parallaxes and the reddening measurements. The reddening due to interstellar dust along the line of sight affects the overall slope and flux level of the observed spectrum, thus influencing both the radius and T_eff measurements. Similarly, the precision on the Gaia parallax directly reflects that in the radius and hence in the white dwarf mass.

Allowing a |$3\, \sigma$| variation for both the parallax and the reddening, the observed flux level of a system could be reproduced by a combination of high reddening (i.e. more absorption along the line of sight) and a large parallax (i.e. moving the object closer to the observer) and vice versa. This degeneracy cannot be broken from the sole analysis of the HST data since they do not extend to the wavelength range where the signature of interstellar dust absorption could be detected as a bump at |$\simeq 2175\,$| Å, from which the colour excess can be estimated.

Ideally, the spectroscopic fitting procedure described in the previous section could account for this correlation by allowing both the distance and the reddening to vary according to suitable priors. In practice, this approach would require including in our fitting procedure the probability density function we used to compute the distance to each system in Section 2.1.1. Since this is not straightforward, we preferred to use an alternative method and estimate the systematic uncertainty on our mass measurements employing a Monte Carlo approach. We used a χ² minimization routine to fit each spectrum 5000 times. During each execution, the models were scaled assuming a distance drawn randomly from the probability density function used to compute the distance to each system in Section 2.1.1. Similarly, the models were also corrected for the reddening, assuming a colour excess drawn randomly from a normal distribution, centred on the E(B − V) reported in Table 1 and weighted according to its uncertainty. Keeping the distance and the reddening fixed to the values drawn from these distributions, we allowed as free parameters the white dwarf |$\rm{T_\mathrm{eff}}$| and log (g) and the second continuum component (as defined in the previous Section). To speed up the calculation, we masked the emission lines.

From the best-fitting parameters obtained from each of the 5000 executions, we derived the posterior distribution of |$\rm{T_\mathrm{eff}}$| and log (g), which provided the estimates of the systematic uncertainties related to the accuracy on the distance and the reddening. We compared these systematics with the statistical uncertainties derived in the previous Section and assumed as final value the maximum of the two.

In the next few years, the upcoming Gaia data releases will improve the accuracy on the parallaxes, and thereby that on the distances to CVs, and will allow reconstruction of more detailed 3D reddening maps. Eventually, these will make it possible to reduce the uncertainties on the white dwarf parameters to the level of the statistical ones.

3.3 Radial velocity measurements from optical spectra

Among the different lines arising from the secondary photosphere listed in Section 2.2, the K i (⁠|$12\, 432/12\, 522$| Å) lines are the strongest and the only ones visible in the spectra of all three systems. The Na i doublet is only visible in the spectra of IR Com but it is contaminated by the residual of the telluric line removal. Therefore, we decided not to include it in the following analysis. We used the Mg ii absorption feature and the K i lines to track the reflex motion of the white dwarf and the donor, respectively. Our data were characterized by a relatively low SNR (≃10–20 in the UVB and ≃5 in the NIR) and, moreover, the NIR spectra showed strong contamination from the residuals from the sky lines subtraction. Therefore, to achieve more robust radial velocity measurements, we fit all the spectra of each object simultaneously (as done e.g. by Parsons et al. 2012 and Pala et al. 2019).

Figure 6.

For each system, the trailed spectra for the Mg ii line (4481 Å, left-hand panel) and the K i line (⁠|$12\, 522$| Å, right-hand panel) are shown. Overplotted are the best-fitting models (red) along with their uncertainties (light blue).

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By assuming the same mass–radius relationship as in Section 3.1.1 and using equation (5), we measured the masses of the three CV white dwarfs, which result in |$0.76^{+0.06}_{-0.07}$|⁠, |$0.95\pm 0.04$|⁠, and |$0.91^{+0.14}_{-0.20}\, \mathrm{M_\odot }$|⁠, for AX For, IR Com, and V1108 Her, respectively (Table 6). In Section 4.1.3, we discuss these results in comparison with the masses obtained from the analysis of the ultraviolet spectra.

The accuracy we achieved on these measurements is directly related to the quality of the data. An optimal sampling of the orbital period is crucial to precisely measure radial velocities. This can be seen by comparing the results for IR Com with those for AX For and V1108 Her. The data of the former are almost evenly distributed along the orbital period and allow us to obtain accuracy of the order ≃7 per cent. However, the observations of AX For were affected by clouds while those of V1108 Her were contaminated by the presence of a close background star. In both cases, we were forced to reject some spectra, resulting in poor orbital sampling and larger uncertainties.

3.4 Summary on the mass measurements

The results of our fitting procedures are summarized in Table 5. In the case of the white dwarf in AX For, from the analysis of the ultraviolet data, we derived a smaller radius (⁠|$\simeq 0.0142\, \mathrm{R_\odot }$|⁠) than the one estimated from the gravitational redshift (⁠|$\simeq 0.0190\, \mathrm{R_\odot }$|⁠). We can thus roughly estimate that at the time of the HST observations |$\simeq 77\,$| per cent of the white dwarf surface was still heated by the recent outburst and, therefore, we assumed as final measurements for its mass and radius those obtained from the gravitational redshift in Section 3.3.