© ESO 2020

1. Introduction

Of the more than 1200 cataclysmic variables in the final 2016 edition of the Ritter & Kolb (2003) catalog, 114 are confirmed polars (or AM Herculis binaries), which contain a late-type main sequence star and an accreting magnetic white dwarf in synchronous rotation. The name “polar” was coined by Krzeminski & Serkowski (1977) to describe the high degree of circular polarization, which became one of the hallmarks of the class. Another is the large portion of the bolometric luminosity emitted in high states of accretion in form of soft and hard X-ray emission, which led to the discovery of the majority of the known systems. Many individual polars are characterized by idiosyncrasies, which distinguish them from their peers and provide special insight into the physics of polars. Unresolved questions relate, for example, to the physics of accretion (Bonnet-Bidaud et al. 2000, 2015; Busschaert et al. 2015), various aspects of close-binary evolution (Webbink & Wickramasinghe 2002; Liebert et al. 2005; Knigge et al. 2011), and the generation and structure of the magnetic field of the white dwarf (WD; Beuermann et al. 2007; Wickramasinghe et al. 2014; Ferrario et al. 2015).

Our optical programs for identifying high-galactic latitude ROSAT X-ray sources (Thomas et al. 1998; Beuermann et al. 1999; Schwope et al. 2002) have led to the discovery of 27 new polars. Twenty sources have been described in previous publications. In this series of three papers, we present results on the remaining seven. Paper I (Beuermann et al. 2017) describes V358 Aqr (=RX J2316–05), a system that experiences giant flares on its secondary star. Here we present a comprehensive analysis of the eclipsing polar HY Eri (=RX J0501–03) based on data collected over three decades. Our early conference paper (Burwitz et al. 1999) represents the only previous account of the system in the literature. The third paper of this series will contain shorter analyses of RX J0154−59, RX J0600−27, RX J0859+05, RX J0953+14, and RX J1002−19, of which three have not been addressed previously either.

2. Observations

2.1. X-ray data

HY Eri, located at RA(2000) = ${05}^{\mathrm{h}}{01}^{\mathrm{m}}46\stackrel{″}{.}4$, Dec(2000) = −03° 59′20″ (l, b = 203.5, −26.1) was discovered as a very soft X-ray source in the RASS (Boller et al. 2016)¹ and identified by us spectroscopically with an eclipsing polar (Beuermann & Thomas 1993; Beuermann et al. 1999; Burwitz et al. 1999). Follow-up pointed ROSAT observations were performed 1992 and 1993 with the Position Sensitive Proportional Counter (PSPC) as the detector and 1995 and 1996 with the High Resolution Imager (HRI). These data were originally published by Burwitz et al. (1999) and reanalyzed for the present study. We also analyzed the previously unpublished data taken in 2002 with XMM-Newton equipped with the EPIC camera. On all occasions, HY Eri was encountered with an X-ray flux that corresponds to a high or near high state² (Table 1).

2.2. Optical photometry

Orbital BVRI light curves and V-band eclipse light curves were measured in 1994 and 1996 with the ESO/Dutch 0.9 m telescope (Burwitz et al. 1999). Extensive white-light (WL) photometry, performed between 2010 and 2019 with the 1.2 m MONET/N and MONET/S telescopes at the McDonald Observatory and the South African Astrophysical Observatory, respectively, allowed us to establish an alias-free long-term ephemeris. Seven-color grizJHK photometry was performed with the MPG/ESO 2.2 m telescope equipped with the GROND³ photometer in 2016, 2017, and 2018. A log of the observations is provided in Table 2. We measured magnitudes relative to the dM1-2 star SDSS 050146.02−040042.2 (referred to as C1), which is located 43″ E and 4″ N of HY Eri and has Sloan AB magnitudes g = 16.91, r = 16.38, i = 16.23, and z = 16.17. Its color, g − i = 0.68, is similar to the low-state Sloan color of HY Eri, g − i = 0.71. HY Eri is separated by only $6\stackrel{″}{.}2$ from the center of a galaxy with Sloan r = 18.44. All accepted eclipse light curves were taken in sufficiently good seeing to escape spillover from the galaxy.

2.3. Optical spectroscopy

Follow-up time-resolved spectrophotometric observations of HY Eri in its high state were performed in 1993 and 1995, using the ESO 1.5 m telescope with the Boller & Chivens spectrograph and the ESO/MPG 2.2 m telescope with EFOSC2, respectively. In the latter run, grisms G1 and G3 yielded low- and medium-resolution spectra with FWHM resolutions of 25 Å and 8 Å that covered the entire optical band and the blue band, respectively. Using the photometrically established eclipse ephemeris, a 10 min exposure in the near-total eclipse was taken with the ESO/VLT UT1 equipped with FORS 1 on 20 November 2000. Grating G300I provided coverage of the red part of the spectrum, which is dominated by the secondary star of HY Eri. The magnetic nature of the WD was studied in 2008 by phase-resolved low-resolution circular spectropolarimetry performed with the ESO/VLT UT2 and FORS1. Grism G300V provided coverage from 3800 to 9200 Å. Table 3 lists the wavelength ranges, number of spectra, exposure times, and total times spent on source.

2.4. Synthetic white light photometry

As described in Paper I, we performed synthetic photometry in order to tie the WL measurements obtained with the MONET telescopes into the standard ugriz system. We defined a MONET-specific WL AB magnitude w, which has its pivot wavelength λ_piv = 6379 Å in the red part of the Sloan r band. For a star with the colors of comparison star C1, the color difference is w − r = +0.07. Hence, unity relative WL flux corresponds to r = 16.38 and w = 16.45. For WL measurements of other stars, w is a measured quantity and is related to Sloan r by r = w − (w − r)_syn. For a wide range of incident spectra, the synthetic color |(w − r)_syn|≲0.1. Hence, w ≃ r is typically correct within 0.1 mag, except for very red stars.

3. Optical light curves

3.1. Orbital light curves

In Fig. 1, we show the V-band and WL light curves in the high states of 11 January 1996 and 19 January 2010, respectively, the WL light curve in the intermediate state of 14−18 November 2010, and the riz low-state light curves of 24−25 November 2017. Orbital phase ϕ = 0 is defined by the eclipse ephemeris provided in Sect. 3.3. HY Eri reached orbital maxima of V = 16.8 and w = 17.1 in the high states and w = 18.7 in the intermediate state. The 2017 peak magnitudes were z = 19.1 and i = 19.7, while r stayed at 21 throughout the orbit outside eclipse. In all states, the light curves exhibit the signatures of emission from two accretion regions, being shaped by cyclotron beaming. The same holds for the light curves in the right panel of Fig. 6. Borrowing from the insight provided by the low-state spectropolarimetry (Sect. 6.1), we identify, for instance, the double-humped z-band light curve in the lower left panel of Fig. 1 as cyclotron emission in the 4th harmonic from two accretion regions best viewed at phases φ ≃ 0.35 and 0.85. The light curves in WL are less easily interpreted because of the lack of wavelength resolution. We loosely refer to the emission regions best seen around ϕ = 0.85 and ϕ ≃ 0.35 as “pole 1” and “pole 2” or “spot 1” and “spot 2”, respectively. In the high state, the emissions from both poles become an inextricable conglomerate. The available evidence suggests that HY Eri is a permanent two-pole emitter.

Fig. 1. Left, from top: orbital light curves of HY Eri in the 1996 and January 2010 high states, November 2010 intermediate, and 2017 low state, binned to ∼2 min time resolution. Right, from top: binned ROSAT X-ray light curves measured with either the PSPC or the HRI as detectors and XMM-Newton light curve measured with the MOS and pn detectors of the EPIC camera.

3.2. Eclipse light curves

We collected a total of 41 eclipses of the hot spot on the WD by the secondary star, 13 in various high states, eight in intermediate states, and 20 in low states. Ingress and egress of the hot spot take place in typically less than 20 s. In Fig. 2, we show the eclipse light curves in the three states at the original time resolutions of 13 s (exposure and readout) for the high and intermediate states and and 22 s for the low state. A characteristic feature of the high state is the delayed eclipse of the accretion stream. This component disappears, when the accretion rate drops.

Fig. 2. Left: eclipse light curves in the high, intermediate, and in the low state, the former two shifted upward to avoid overlap. Phases are from Eq. (1) for the high and intermediate states and from and Eq. (2) for the low state. Right: O − C diagram for the deviations of the mid-eclipse times from the linear ephemeris of Eq. (1), showing the change in orbital period.

The eclipse was modeled by the occultation of a circular disk of uniform surface brightness, which represents either the WD or the hot spot on the WD. The parameters of the fit are the mid-eclipse time, the FWHM, and the duration of ingress or egress. We improved the statistics and reduced the timing error in the low state, when the star became as faint as w = 20.7 outside eclipse, by fitting up to n = 4 eclipses on a barycentric time scale if taken nearby in time. The resulting mid-eclipse times are listed in Table 4. In the cases with n >  1, the cycle number of the best-defined eclipse is quoted. The measured FWHM of the eclipse is the same for the different accretion states with a mean value of Δ t_ecl = 910.6 ± 1.5 s or Δ ϕ = 0.08859 ± 0.00015 in phase units. This is the longest relative eclipse width of all polars. We did not detect the ingress and egress of the WD photosphere because our WL observations are dominated by cyclotron emission and the measurements in the GROND gr filters lacked the required time resolution. The quoted mid-eclipse times refer to the hot spot on the WD and may deviate from true inferior conjunction of the secondary by up to ∼30 s or ∼0.003 in phase.

The relative WL flux in the totality is the same in the high, intermediate, and low states, with a mean of 0.0060 ± 0.0005 or w = 22.0 ± 0.1. Using a color correction w − r ≃ −0.3, appropriate for the secondary star, we find r ≃ 22.3, which compares favorably with the result of our spectrophotometry in eclipse reported Sect. 5.2. Hence, the residual WL flux in eclipse is largely that of the secondary star.

3.3. Eclipse ephemeris

We corrected the UTC eclipse times to Barycentric Dynamical Time (TDB), using the tool provided by Eastman et al. (2010)⁴, which accounts also for the leap seconds. The complete set of eclipse times is presented in Table 4. The 2010 and 2011 data and the scanty earlier data define the alias-free linear ephemeris

$$\begin{array}{c}\hfill T_{\mathrm{ecl}}=\mathrm{BJD}(\mathrm{TDB})2455865.87081(1)+0.118969076(2)E,\end{array}$$(1)

with χ² = 19.6 for 18 degrees of freedom (solid line, cycle numbers E ≤ 0). The O − C offsets from the ephemeris of Eq. (1) are displayed in Fig. 2, right panel. This ephemeris became increasingly invalid after 2011 and the more recent data are well represented by a cubic ephemeris for E >  −6000 (solid curve). Currently, O − C relative to the linear ephemeris of Eq. (1) has exceeded 2 min, which is much too large to be explained by wandering of the accretion spot. The mid-eclipse times of 2017 to 2019 follow the linear ephemeris

$$\begin{array}{c}\hfill T_{\mathrm{ecl}}=\mathrm{BJD}(\mathrm{TDB})2455865.86951(14)+0.118969198(8)E,\end{array}$$(2)

implying a period change relative to Eq. (1) of 10.5 ± 0.7 ms. The mean rate of the period variation between 2011 and 2018 is Ṗ ≃ 5 × 10⁻¹¹ ss⁻¹. The period change started approximately, when the system entered a prolonged low state after 2011. This is likely a coincidence, however, because it was also in a low state in 2008 and temporarily in an intermediate state in 2016.

4. X-ray light curves and spectra

The ROSAT soft X-ray light curves taken between 1990 and 1996 (Burwitz et al. 1999) show a structured bright phase that extends from ϕ ≃ 0.4 to ϕ ≃ 0.8 with low-level emission over the remainder of the orbit. Binned versions of these data are shown in Fig. 1, right panels. We also included the previously unpublished light curve measured with XMM-Newton and the MOS and pn detectors of the EPIC camera in 2002. The lower right panel of Fig. 1 shows the mean count rates of the two MOS detectors and the pn-detector, respectively, with the former adjusted by a factor of 3.5 upward. The X-ray bright phases in the ROSAT and XMM-Newton light curves show some similarity with the WL optical light curve of November 2010, suggesting that the observed intense X-ray emission originates from pole 1, phase-modulated by rotation of the WD and possibly by internal absorption. The low-level emission around ϕ = 0.3 may stem from pole 2. The bright phase reached count rates around 1.0 PSPC cts s⁻¹, 0.4 HRI cts s⁻¹, and 1 EPIC-pn cts s⁻¹, suggesting that HY Eri was in some form of high or intermediate-to-high state during these observations.

The ROSAT and XMM-Newton spectra (not shown) are characterized by an intense soft X-ray and an underlying hard X-ray component, of which the latter is well covered only in the XMM-Newton run. We fitted the bright-phase spectra with the sum of a blackbody of temperature kT_bb and a thermal component with the temperature fixed at 10 keV, both attenuated by a column density N_H of cold interstellar matter of solar composition⁵. The fit parameters are listed in Table 5. The values of kT_bb and N_H differ substantially, indicating either true variability or systematic uncertainties caused by the poor energy resolution of the ROSAT PSPC, the lack of spectral coverage of the XMM-Newton detectors below 0.2 keV, or the inadequacy of fitting a multi-temperature source by a single blackbody (Beuermann et al. 2012). In any case, HY Eri is more strongly absorbed than other polars. For the combined PSPC fit to the 1992 and 1993 data, N_H exceeds the total galactic column density N_H, gal ≃ 5.2 × 10²⁰ H-atoms cm⁻² at the position of HY Eri (Hi4PI Collaboration 2016)⁶. The galactic extinction at the position of HY Eri, A_V = 0.172 (Schlegel et al. 1998), and the N_H − A_V conversion factor of Predehl & Schmitt (1995) yield N_H ≃ 3 × 10²⁰ H-atoms cm⁻², which corresponds to the galactic dust layer. Forcing the fits to this value, the ROSAT and XMM-Newton fits yield similar blackbody temperatures and a bolometric soft X-ray flux of F_X ≃ 1.5 × 10⁻¹¹ erg cm⁻² s⁻¹.

5. Optical spectroscopy and spectropolarimetry

5.1. High-state spectra

In Fig. 3, we show the mean low-resolution spectra taken on 13–17 December 1993 and 14 November 1995, when HY Eri was in high states. They are characterized by a blue continuum, strong Balmer, He I, and He II emission lines, the Balmer jump in emission, and weak broad cyclotron lines at the red end. Phase-resolved radial velocities were measured from medium-resolution blue spectra taken on 15 November 1995 (not shown). The Balmer and HeII emission lines were single peaked with asymmetric bases, extending to beyond ±1000 km s⁻¹. We measured the mean central wavelengths of the broad components and the positions of the line peaks of Hβ, Hγ, and HeIIλ4686. The former has a velocity amplitude of 520 ± 24 km s⁻¹ and reaches maximum positive radial velocity at ϕ = 0.91 ± 0.02. This phasing is consistent with the plasma motion in the magnetically guided stream that leads to pole 1 and away from the observer at an azimuth of ψ ∼ 35°, measured from the line connecting the two stars. The line peak displays a small radial velocity with zero crossing near ϕ = 0.80. This component could arise from the ballistic accretion stream near L₁.

Fig. 3. Top: mean flux-calibrated low-resolution spectra of HY Eri in the high states of 1993 and 1995. For comparison, the eclipse spectrum of 2000 (blue curve) and the mean out-of-eclipse spectrum in the 2008 low state (red curve) are added on the same ordinate scale. Bottom: mean radial velocities of the broad emission lines of Hβ and HeIIλ4686 (open circles) and of the line peaks (cyan dots) derived from medium-resolution spectra taken on 15 November 1995.

5.2. Spectrum of the secondary star in eclipse

On 20 November 2000, we acquired a spectrum of the secondary star during the WD eclipse, using the ESO VLT UT1 with FORS1 and grism G300I (Table 3). The 600 s exposure was preceded by three and followed by five 30 s exposures. The run started shortly after the ingress of the accretion spot into eclipse and extended until after its egress. The dotted vertical lines in (Fig. 4, top panel) indicate the duration of the eclipse. The 600 s exposure covered the phase interval ϕ = −0.0149 to 0.0435, beginning after the stream component subsided and ending just before the spot starts to reappear at ϕ = 0.0438. The figure shows the light curves of the Hα line flux and of the continuum near Hα integrated over 30 Å. The Hα emission stays finite in the eclipse.

Fig. 4. Top: eclipse light curve of the Hα emission line flux in erg cm−2 s−1 and of the underlying continuum flux observed on 20 November 2000. Center: eclipse spectrum (black curve) thermal hydrogen spectrum adjusted to fit the Hα line flux (blue or red curve, see text). The ordinate is in units of 10−16 erg cm−2 s−1 Å−1. Bottom: difference spectra on the same color coding fitted by a dM6 star (red) and a dM5 star (blue, shifted upward by 0.05 units).

The center panel of Fig. 4 shows the eclipse spectrum dereddened with the galactic extinction A_V = 0.172 (Schlegel et al. 1998), where we have assumed that HY Eri is located outside the principal dust layer of the galactic disk (Jones et al. 2011). The secondary has a dereddened AB magnitude i = 20.99 with an estimated systematic error of 0.10 mag from the absolute flux calibration of the spectrophotometry. The spectrum shows the TiO features characteristic of a late dM star and a strong Hα emission line with a line flux of 6.8 × 10⁻¹⁶ erg cm⁻² s⁻¹. The line is centered approximately at the rest wavelength and has a velocity dispersion of ∼1000 km s⁻¹. It may arise from a tenuous uneclipsed section of the accretion stream. Regardless of its origin, the line emission will be accompanied by a thermal continuum that we need to define and subtract before a spectral type can be assigned to the secondary star. To this end, we calculated spectra of an isothermal hydrogen plasma of finite optical depth, added line broadening, and normalized the spectra to fit the observed Hα line flux. The free parameters of the model are the electron temperature T_e, the pressure P, and the geometrical thickness x of the emitting plasma. In the center panel of Fig. 4, we show two examples that bracket the permitted range of the flux of the sought-after continuum. The blue spectrum for T_e = 10 000 K, P = 10 dyne cm⁻², x = 10⁸ cm features a low thermal continuum and the red curve for T_e = 20 000 K, P = 200 dyne cm⁻², x = 10⁸ cm a high one. The bottom panel of Fig. 4 shows the observed spectrum with either one of the model spectra subtracted. Employing a set of dereddened spectra of SDSS dM stars, we find that the observed spectrum corrected with the low thermal continuum is best fitted with the dM5 star SDSS J101639.10+240814.2 adjusted by a factor of 287 (rms spectral flux deviation 0.0055 in the ordinate unit of Fig. 4, bottom panel). The corresponding spectrum for the high thermal continuum is equally well fitted with the dM6 star SDSS J155653.99+093656.5 adjusted by a factor of 289 (rms deviation 0.0053). The two cases tap the full range of thermal continua permitted by the eclipse spectrum and spectral types outside the dM5–6 slot quickly fail to provide an acceptable fit. For the dM5 case, the dereddened spectrum of the secondary corrected for the small thermal contribution has i = 21.03 and colors r − i = 1.73 and i − z = 0.95. For the dM6 case with its larger thermal component, we find i = 21.22 with r − i = 2.01 and i − z = 1.10⁷. The angular radius R₂/d of the secondary star and its brightness are related by the surface brightness

$$\begin{array}{c}\hfill S=m+5\log \left[(R_2/d)(10\mathrm{pc}/R_{\odot })\right],\end{array}$$(3)

where m is the magnitude of the star in the selected band, R₂ its radius, and d its distance. We calibrated the surface brightness S_i in the i-band as a function of color, using the extensive data set of Mann et al. (2015) that combines Sloan griz photometry and stellar parameters. The desired relations are

$$\begin{array}{cc}\hfill S_i& =5.34+1.554(r-i)\mathrm{with}\sigma (S_i)=0.086\hfill \end{array}$$(4)

$$\begin{array}{cc}\hfill S_i& =5.33+2.829(i-z)\mathrm{with}\sigma (S_i)=0.072,\hfill \end{array}$$(5)

valid for r − i = 1.0 − 2.6 and i − z = 0.6 − 1.3, respectively. The quoted standard deviations describe the average spread of S_i around the fit. With the colors of the secondary star quoted above, we obtained mean values from Eqs. (4) and (5) of S_i(dM5) = 8.02 and S_i(dM6) = 8.44. From Eq. (3), the distance d in pc is given by

$$\begin{array}{c}\hfill \log d=(i-S_i)/5+1+\log (f_{\mathrm{back}}{R}_2/R_{\odot })\end{array}$$(6)

where i is the magnitude of the continuum-corrected eclipse spectrum, S_i the respective surface brightness, R₂ the volume equivalent radius of the Roche lobe, and f_back ≃ 0.961 the reduction factor for the backside view of the lobe⁸. The dM5–dM6 differences in S_i and in the i-band magnitude compensate in part, leading to similar distances with a ratio d_dM5/d_dM6 = 1.11. We employ Eq. (6) in Sect. 7.5, using the radii of the secondary star derived from our dynamic models of HY Eri. The error in the mean dM5–dM6 distance includes the ±0.10 mag uncertainty in the flux calibration, the ±0.08 mag scatter in S_i, and half the difference of the dM5 and dM6 distances. Added quadratically, the error in d is ±8.1% plus any error that arises from R₂.

We estimated the effective temperature of the secondary star from the color dependence of T_eff in the data of Mann et al. (2015), which yields T_eff ≃ 3070 K and 2900 K for the dM5 and the dM6 case, respectively. With the best-fit system parameters of Sect. 7.5, the luminosity of the secondary star becomes 2.6 − 2.0 × 10³¹ erg s⁻¹ for a spectral type range of dM5−6.

6. Circular spectropolarimetry

We studied the magnetic field of the WD in HY Eri by phase-resolved circular spectropolarimetry performed on 31 December 2008, when the system was accreting at a low level. A total of 40 sets of ordinary and extraordinary spectra were obtained with the ESO VLT UT2 equipped with FORS1 and GrismG300V, covering two consecutive orbital periods (Table 3). The pipeline reduction combines two spectra each for two positions of the λ/4 plate in order to account for possible cross-talk from linear polarization. This procedure yielded set 1 of 20 calibrated intensity and circular polarization spectra, each covering a phase interval Δϕ = 0.10. Spectral features faithfully repeated in both orbits and we phasefolded the intensity and polarization spectra for further analysis. In case of the intensity spectra, the phase resolution can be improved to Δϕ = 0.05 by adding the individual ordinary and extraordinary spectra to form set 2 of 40 provisionally calibrated intensity spectra. Set 1 was employed for a quantitative analysis of the spectral flux, set 2 for tracing the motion of the cyclotron line peaks and measuring the Hα radial velocities. Both were corrected for the secondary star, using the dM5 representation of the eclipse spectrum of Fig. 4 extrapolated into the blue spectral region and set 1 was dereddened with A_V = 0.172 (Schlegel et al. 1998). Hα emission is much weaker than in the 2000 eclipse spectroscopy of Sect. 5.2 and we do not correct for the probably tiny contribution of the associated thermal continuum. Grayscale representations of the corrected phasefolded set-1 intensity and circular polarization spectra are shown in Fig. 5, repeated twice for visual continuity. Rows #1, #11, and #21 represent the eclipse. The orbital mean spectrum outside eclipse (rows #3−10, ϕ = 0.14 − 0.91) is shown in the left panel of Fig. 6 (red curve); an appropriate model spectrum of the magnetic WD is added for comparison (black curve). The two blue circles are the mean out-of-eclipse g and r-band fluxes of our 2017 and 2018 photometry, corrected for the secondary. They show that the 2008, 2017, and 2018 observations were performed in similar states of low accretion. The right panels of Fig. 6 show the light curves for the blue and red wavelength intervals extracted from the set 1 spectra. The red ones (7400 − 7600 Å and 7900 − 8100 Å) are shaped by the beamed optically thin cyclotron emission of poles 1 and 2. The blue one (4000 − 4800 Å) represents the sum of the photospheric and spot emissions of the WD. Disentangling the two components proved infeasible. The Hα line emission is discussed in Sect. 7.3.

Fig. 5. Left: intensity spectra of the 2008 spectropolarimetry with the contribution of the secondary star subtracted. The mean spectra of two orbits are shown twice for visual continuity. Two systems of cyclotron lines are visible that originate near two poles of similar field strength. Center: circular polarization spectra with white indicating positive and black negative polarization. Right: maxima of the cyclotron lines of both poles (circles, triangles), extrema of the circular polarization (plus signs, crosses), and best-fit line positions as functions of orbital phase (red curves).

Fig. 6. Left: mean bright-phase dereddened intensity spectrum of the 2008 spectropolarimetry with the contribution of the secondary star subtracted (red curve). For comparison, the appropriate model spectrum of a magnetic WD is shown (black curve). The mean fluxes of the November 2017 and February 2018 gr photometry are included as the two open blue circles. The red numbers indicate the cyclotron harmonics. Right: light curves for selected wavelength intervals derived from spectral set 1 before phasefolding, but after subtraction of the secondary star. The individual panels show the quasi-B band flux, the cyclotron beaming of the pole-1 and pole-2 emissions, and the Hα line flux (in arbitrary units).

6.1. Cyclotron spectroscopy

The cyclotron lines of pole 1 are visible from ϕ = 0.6 − 1.1 with positive circular polarization and those of pole 2 in the remainder of the orbit with negative circular polarization. Overlaps between the two line systems occur in rows #3 (ϕ = 0.14 − 0.23) and and #7 (ϕ = 0.54 − 0.63), where the spectra show the signatures of both poles. We extracted the cyclotron lines from the individual intensity spectra by removing the underlying WD continuum. The cyclotron line profiles were then subjected to least squares fits using the theory of Chanmugam & Dulk (1981) for an isothermal plasma. The free parameters of the model are the plasma temperature kT, the field strength B, the viewing angle θ against the field direction, the thickness parameter Λ and a remnant optically thick continuum represented by a second-order polynomial (dashed lines in Fig. 7). The thickness parameter Λ of cyclotron theory is related to the column density x_s of the cooling region by Λ = 4πex_s/μ_em_uB, where e is the elementary charge, μ_e the number of electrons per nucleon in the plasma, m_u is the atomic mass unit, and B the field strength.

Fig. 7. Left: cyclotron spectra of poles 1 and 2 (red curves) least-squares fitted with constant-temperature models for the plasma parameters listed in Table 6 (black curves). Right panels: magnetic geometry of the WD for case A and case B accretion (see text). The secondary star is located to the left. Angles are measured between the dots on the circumference. Dipolar field lines are included as red curves.

Because of trade-off effects between kT and log Λ, reliable values of these two parameters can not be determined without additional information. We took recourse to the results of the two-fluid radiation-hydrodynamic cooling theory of Fischer & Beuermann (2001) and explain our approach for the case of the pole-2 spectrum of row #6 in the lower left panel of Fig. 7. Good fits to that spectrum were obtained along a narrow valley in the Λ − kT plane that follows log Λ = 6.64 − 4.73 log(kT) and extends to parameters quite inappropriate for a post-shock region dominated by cyclotron cooling. Cooling theory provides a second relation between log Λ and log(kT) that runs nearly orthogonal to that of the line fits. For the case of the #6 spectrum, it reads log Λ = 4.97 + 2.50 log(kT)⁹. The intersection of the two relations defines the most probable values of log Λ and kT, yielding kT = 1.70 keV and log Λ = 5.55. Table 6 lists the corresponding fits to all spectra that can uniquely be assigned to either pole 1 or pole 2. The emerging picture is that of two emission regions with similar field strengths B_sp ≃ 27.5 MG and 28.8 MG, thickness parameters log Λ ≃ 5.8 and 5.5, and temperatures of kT ≃ 2.2 keV and 1.7 keV for poles 1 and 2, respectively. The third parameter derived from the cyclotron fits is the mean viewing angle ⟨θ⟩ between the line of sight and the direction of the accreting field line averaged over the spot. Closest approach to the field line occurs for pole 1 at ϕ ≃ 0.87 in row #10 and for pole 2 at ϕ ≃ 0.33 in rows #4 and #5 with minimum viewing angles of ⟨θ_min, 1⟩≃49° and ⟨θ_min, 2⟩≃62° for spots 1 and 2, respectively. Since the cyclotron lines widen and weaken rapidly with decreasing θ, the quoted angles may somewhat overestimate the true mean values.

Table 6 also lists the mass flow densities ṁ that are delivered also by two-fluid cooling theory. They fall far below the ∼1 g cm⁻² s⁻¹ of a bremsstrahlung-dominated emission region and are only about an order of magnitude away from the transition to the non-hydrodynamic regime of the bombardment solution (Woelk & Beuermann 1992; Fischer & Beuermann 2001).

Complementary information on θ is obtained from the orbital motion of the cyclotron line peaks and of the circular polarization extrema. The peak wavelengths measured from the spectra of sets 2 and 1, respectively, are shown in the right panel of Fig. 5. Near ϕ = 0.1 and 0.6, both line systems overlap and are difficult to disentangle. As in Paper I, the motion of the cyclotron lines was modeled, using a parameterized form of the frequencies of optically thin harmonics in units of the cyclotron frequency as functions of kT and θ. We defined the field vectors in the two spots as B_sp, 1 and B_sp, 2 and obtained field strengths and directions by least-squares fitting the phase-dependent motion of the line peaks of the fifth and sixth harmonics (red curves in Fig. 5). As input we used the plasma temperatures of Table 6 and an inclination of i = 80° from Table 9. The results are presented in Table 7, where we list the field strengths B, the azimuth angles ψ_f, and the colatitudes δ_f of the accreting field lines. The results are quoted for two accretion geometries with pole 1 either in the “southern” hemisphere below the orbital plane (1S) or in the “northern” one above it (1N). Note that the fit does not provide information on the location of the spots and the orientation of the magnetic axis. The angle between the field vectors B_sp, 1 and B_sp, 1 is not far from 180°, at least in the 1N–2S case. Combined with the fact, that both spots display circular polarization of opposite sign, the data suggest a field structure that is dominated by a dipole and possibly octupole rather than a quadrupole.

6.2. Zeeman spectroscopy

The blue continuum in the left panel of Fig. 5 represents the photospheric emission of the WD including a strong spot component. To facilitate the Zeeman analysis, we transformed the set-1 spectra to the rest system of the WD, employing the preferred dynamical model of Sect. 7.5. There is little orbital variation in the Zeeman lines except near the transitions between the visibility of poles 1 and 2. We adopted the averages of rows #8–10 (ϕ = 0.63 − 0.91) or #3–6 (ϕ = 0.14 − 0.52) as representative of the hemispheres that include pole 1 or 2, respectively. The dominant field strengths are 27–28 MG in pole 1 and 29–30 MG in pole 2. The 4000–5250 Å section of the pole-1 spectrum is shown in Fig. 8 (red curve). Unfortunately, the circular polarization spectra, which contain information on the direction of the magnetic field vector B, are too noisy to be of any use, limiting our ability to distinguish between different magnetic field structures that fit the intensity spectra similarly well.

Fig. 8. Left panels: magnetic-field distributions of the WD for the case B multipole model of Table 8 and the phases of best visibility of spots 1S and 2N. The footpoints of the rotational and the magnetic axes are indicated by the black dot and the cross (×), respectively. The magnetic equator and the orbital plane are marked by the dashed and the solid black line, respectively. The individual projected-area elements are marked by tiny dots, selecting or suppressing every second integer field strength. The spot field is marked in green and the viewing-angle selected spot is emphasized in black. The bottom panels show the model field strength vs. the magnetic colatitude ϑ. The red dots mark the spot fields. Right panel: observed pole-1 spectrum (red curve) with the best-fit Zeeman spectrum for the adopted multipole model (black curves). The residuals of the fit are displayed at the bottom with the wavelength sections that are included in the fit highlighted in blue.

Our spectral synthesis program employs an improved set of the model Zeeman spectra calculated by Jordan (1992) and previously used by Euchner et al. (2002, 2005, 2006) and Beuermann et al. (2007). The present version includes the Balmer lines up to Hδ and consists of the log g = 8 intensity spectra for 16 effective temperatures from 8 to 100 kK, integer field strengths B from 1 to 100 MG, and 17 viewing angles θ = 0° to 180°, uniformly distributed in cos θ. The model spectra were calculated for a Stark broadening factor C = 0.1 (Jordan 1992). Interpolation in T_eff and in θ is unproblematic, while interpolation in B is impracticable. The spectra were smoothed to match the observed resolution of 10 Å FWHM. At this resolution, the 1-MG spacing is just about adequate and misfits stay small.

We considered a magnetic model that includes the zonal multipole components of degree ℓ = 1 − 3, that is the aligned dipole, quadrupole, and octupole, inclined by a common angle α against the line of sight. For the relative field strengths r_dip = 1 − r_oct, r_qua, and r_oct, the combined polar field strength is B_p = (1 + r_qua)B₀, with B₀ a scaling factor. We divided the visible disk of the WD into 6561 limb-darkened projected area elements, and collected them into 17n field-strength and viewing-angle bins (k, l), k = 1…n for the nearest integer field strength B_k and l = 1…17 for the nearest cos θ_l value. The unreddened spectral flux at the Earth is

$$\begin{array}{c}\hfill f_{\lambda }(B_0,r_{\mathrm{oct}},\alpha )={(R_{\mathrm{w}}d/d)}^2{\displaystyle \sum _{k=1}^{\mathrm{n}}\sum _{l=1}^{17}{a}_{k,l}{F}_{\lambda }(B_k,T_{k,l},{\theta }_l),}\end{array}$$(7)

where a_k, l is the integrated limb-darkened fractional projected-area of bin (k, l), F_λ(B_k, T_k, l, θ_l) the data bank spectrum for that bin at the interpolated temperature T_k, l, and (R_wd/d)² the dilution factor, with R_wd the WD radius and d the distance. The best values of B₀, r_qua, r_oct, and α were determined in a grid search, the T_k, l and the dilution factor by a formal least squares fit at each grid point. Obtaining a stable fit, requires a severely restricted number of independent temperatures. The spot emits about 2/3 of the blue flux from ∼5% of the area, requiring a minimum of two temperatures, naturally identified as a high spot temperature T_sp and a low photospheric temperature T_ph. Some fits benefit from a minor third component, such as a warm polar cap with T_cap. We extended the fit over the wavelength interval 4000 − 5200 Å, excluding sections around the Balmer emission lines, some HeI lines, and an unidentified line complex around 4300 Å. A formal χ² was calculated for 85 resolution elements of 10 Å width, using relative flux errors of 1.8% for pole 1 and 2.4% for pole 2, measured from the scatter among the set-1 intensity spectra.

6.3. Stark broadening

An accepted theory of Stark broadening in the presence of a magnetic field does not exist. Jordan (1992) opted to equate the broadening of the individual Stark components of a Balmer line to the mean Stark shift of all components multiplied by a factor C ≃ 0.1 and Putney & Jordan (1995) considered values of C = 0.1 and 1.0 for stars with vastly different field strengths. It is necessary, therefore, to consider the appropriate level of the line broadening for a given application. To this end, we adopted an approximate post factum procedure that changes the line strengths, while avoiding recalculation of the data base. We expressed the line profiles in terms of an optical depth τ_λ, setting $F_{\lambda }=F_{\lambda }^{\text{c}}{e}^{-{\tau }_{\lambda }}$, with $F_{\lambda }^{\text{c}}$ the continuum flux. We then replaced F_λ in Eq. (7) by $F_{\lambda }^{\text{new}}=F_{\lambda }^{\text{c}}{e}^{-{\tau }_{\lambda }\eta }$ and included η as an additional free parameter in the grid search. For our best multipole model, we obtained χ² = 85.5 at η = 1 (C = 0.1), the best fit with ${\chi }_{\text{min}}^2=84.8$ was attained at η = 1.25, and the 90% confidence level with ${\chi }_{\text{min}}^2=87.5$ was reached for η = 0.80 and 2.50, with χ² quickly rising at still lower and higher η. Hence, our fit favors line strengths somewhat larger than nominal (C = 0.10, η = 1.0). This result applies to our simultaneous multi-temperature fits to spectral flux and line strengths and may not be generally valid. We adopted η = 1.25 for the present paper and obtained the systematic errors at the 90% confidence level for η = 0.8 − 2.5. The large errors re-emphasize the need for an effort to calculate the Stark shifts of the individual Stark components in a magnetic field.

6.4. Magnetic geometry of the accreting WD

The right-hand panels of Fig. 7 show two selected magnetic WD geometries. For simplicity, both have rotational pole, magnetic pole, and the accretion spots on the same meridian (here the paper plane). The secondary star is located far to the left. The footpoints of the common magnetic axis are displaced from the respective viewing directions by α₁ and α₂, with α₁ + α₂ = 180° −2i, where i is the inclination and α is counted positive away from and negative toward the rotational pole. In case A, both spots are located between magnetic pole and viewing direction and can accrete from the nearby orbital plane. To reach spot 2, the plasma must travel halfway around the WD before it attaches to a near-polar field line. In case B, spot 1 can accrete from the nearby orbital plane. Spot 2, however, is located between magnetic and rotational pole and the field line leads over the rotational pole in the general direction of the secondary star. Although energetically unfavorable, this non-standard path may be active and it is not clear whether the trip over the pole or the travel around the WD should be dismissed as the less likely way to feed spot 2.

In perusing parameter space, we found that all good fits require field strengths larger than the spot field B_sp and are rather insensitive to a lack of small field strengths. We started from a pure dipole model that fits the pole 1 and pole 2 spectra with B₀ = 37.5 and 40.5 MG, respectively. Adding a small quadrupole component, leads to a common B₀ = 38.5 MG. The parameters of this quasi-dipole fit are listed in Table 8. As expected for an inclination of 80° (Sect. 7.5 and Table 9), α₁ + α₂ ≃ 20°, confirming the presence of a common magnetic axis for the separately performed fits. The colatitudes δ_f of the accreting field lines agree reasonably well with those of the 1S−2N geometry in Table 6, considering the uncertainties of about 5°. Superficially, the fit seems close to perfect were it not for the disturbing fact that the geometry probably prevents accretion in both spots. The ribbon-like spots are offset from the respective magnetic pole by ϑ_sp ∼ 50°, the field lines in the spots reach out to only 1.7 R_wd, and both field lines curve away from the orbital plane. Hence, the quasi-dipole model provides no convincing accretion geometry. Increasing the quadrupole component provides no remedy. For r_qua up to ±0.40, none of the seemingly good fits matches the requirements set up by the cyclotron fits. The same holds for moderate octupole components r_oct up to ±0.40, some of which predict “spots” in the form of near-equatorial ribbons connected by tightly closed field lines.

The situation changes fundamentally for larger octupole components with r_oct ≲ −0.45. With decreasing r_oct, the best-fit values of B₀ in the primary minima of poles 1 and 2 converge and coincide for r_oct = −0.77 and r_qua = ∓0.25, respectively. Table 8 lists the fit parameters. As required, α₁ + α₂ ≃ 20° and the spot-averaged colatitudes ⟨δ_f⟩ and viewing angles ⟨θ_f⟩ agree reasonably well with the 1S−2N cyclotron results of Table 6. Both spots are located closer to the magnetic poles than in the quasi-dipole case and the field lines are close to radial with inclinations ⟨β_f⟩≃0, indicating that the field lines reach far out. The multipole model represents a convincing solution, provided the case B path to spot 2 is active. In passing, we note that enforcing case A accretion at spot 2 by increasing its colatitude fails because a corresponding χ² minimum does not exist. Switching hemispheres, the 1N geometry is a mirror image of 1S, but a χ² minimum at the parameters expected for 2S does not exist either. The general caveat holds that a multipole model with tesseral harmonics may provide a different answer.

The two left panels of Fig. 8 show the magnetic field distributions of the WD for the multipole model of Table 8 at the phases of the best visibility of spots 1S and 2N. The yellow and green bands indicate the spot field strengths of 27 and 28 MG for pole 1 and 29 and 30 MG for pole 2 and the black portions the viewing-angle selected spots. Although both aspects belong to the same field model, the spot geometries differ significantly. So do the full ranges of the field strengths over the visible face of the WD (bottom panels, see also Col. 17 of Table 8). The right panel of Fig. 8 shows the pole-1 spectrum (red curve) and the Zeeman fit (superposed black curve), which faithfully reproduces most Zeeman lines in the spectral regions that are free of atomic emission lines. The contributions by the spot and the photosphere+cap are also shown individually. The Zeeman lines are prominent in the spot component because of the small spread in field strength, but are washed out in the photospheric component. The fit to the pole-2 spectrum (not shown) excels at λ >  4500 Å, but is inferior at shorter wavelengths, possibly because it is composed of only nine independent magnetic spectra (Table 8, Col. 17).

6.5. WD parameters and their errors

The multipole Zeeman fit to the observed pole 1 and pole 2 spectra yielded temperatures for the photosphere, cap, and spot of T_ph = 9.2  ±  0.7 kK (Table 8) and T_cap = 14.2  ±  1.0 kK and T_sp = 78 ± 8 kK (Table 9). In most fits, the flux contributed by the cap is a minor entity. The corresponding angular radius of the WD is R_wd/d = (1.15 ± 0.12)×10⁹ cm kpc⁻¹. All quoted errors refer to the 90% confidence level and include besides the statistical also the systematic error caused by the remaining uncertainty in the level of Stark broadening (Sect. 6.3). Quadratically adding the error from an estimated 10% uncertainty in the flux calibration of the spectropolarimetry gives R_wd/d = (1.15 ± 0.14)×10⁹ cm kpc⁻¹. The error budget of R_wd includes in addition the 8.1% uncertainty in the mean dM5–dM6 distance from Sect. 5.2, raising the error of R_wd to 14.1%. Despite its large error, the measured radius proves helpful in determining the system parameters in Sect. 7.5.

The photospheric temperature of the WD in HY Eri is lower than in practically all other well-studied polars (Townsley & Gänsicke 2009). The low temperature is directly related to the low observed equivalent width W_obs of the Zeeman lines in HY Eri. The equivalent width in the theoretical Zeeman spectra has its peak at T_eff = 12 kK and drops rapidly toward lower and higher temperatures. W_obs falls by a factor of 3.2 below the peak value for nominal Stark broadening. Fitting W_obs and the spectral slope simultaneously, requires T_ph significantly below 12 kK and T_sp ≫ 12 kK. The fit deteriorates with rising T_ph and becomes unacceptably bad at 11 kK, even allowing for a variation in the level of Stark broadening.

7. System parameters

7.1. Narrow emission lines as tracers of the motion of the secondary star

Polars feature narrow emission lines of hydrogen, helium and metals that are thought to originate on the irradiated face of secondary star. In some polars, however, the radial velocity amplitudes of individual lines differ. Helium lines with their high ionization potential show lower amplitudes than hydrogen, while the low-ionization near infrared Ca II lines have the greatest and rather stable amplitudes (Schwope et al. 2000, 2011; Schwope & Christensen 2010). Obviously, the distribution of the emission differs between individual lines, with the helium lines probably originating, in part, from structures outside the chromosphere of the star, such as coronal prominences. Modeling is straightforward as long as the emission originates from locations geometrically close to the surface of the star, which appears to be the case for the low-ionization metal lines and, in some polars, for hydrogen lines. Describing the line emission over the secondary star requires either a dedicated theoretical model or an empirical ansatz that derives the distribution from unfolding highly resolved observed line profiles.

7.2. Models of the irradiated secondary star

We calculated the radial velocity amplitude $K_2^{\prime }$ of the narrow emission line, considering a Roche-lobe filling star that is irradiated by a source at the position of the WD. Each surface element receives an incident flux f_in ∝ cos ϑ/δ², with ϑ the angle of incidence and δ the distance from the source. In response, it emits a line flux f_line that varies with f_in, but may depend on additional parameters. For a given model, we calculated synthetic emission line spectra and determined the lever arm of the line emission region $a_{\text{irr}}(q)=a_2{K}_2^{\prime }/K_2$ as a function of the mass ratio q = M₂/M₁, with a₂ = 1/(1 + q). Below, we quote polynomial approximations for a_irr(q)¹⁰.

The irradiation model of Beuermann & Thomas (1990; henceforth BT90), equates the line intensity emitted from a surface element to a fraction of the total incident flux, f_line ∝ cos ϑ/δ². The emitted intensity drops from a maximum at L₁, where ϑ ≃ 40°, down to zero at the terminator of the irradiated region, where ϑ = 90°. Hence, BT90 favors emission from regions near L₁. The modified version BT90m uses f_line ∝ (cos ϑ)^m/δ², with m a heuristic free parameter. This modification was motivated by the study of irradiated WD atmospheres by König et al. (2006, their Sect. 4.2), who found that the narrow emission cores of Lyα increased drastically when ϑ approached 90° and the incident energy was deposited increasingly higher up in the atmosphere. There is no simple way to relate m to physics, however. An entirely different approach was taken in Beuermann & Reinsch (2008, their Sects. 5.4, 6.1, and Fig. 10), where we determined the distribution of the Ca IIλ8498 emission as a function of ϑ empirically by unfolding the high-resolution line profiles of the intermediate polar EX Hya. A parameterized form of the Ca II emission model was implemented in our model BR08. It provides an internally consistent description of the emission of Ca IIλ8498 and numerous other metal lines that share the motion of Ca II. The model gives larger weight to surface elements near the terminator, equivalent to moderate limb brightening. To put the models into perspective, we note that BR08 corresponds approximately to BT90m with m ≃ 0.3. BT90 and BR08 bracket about the full range of possible irradiation scenarios of the atmosphere of the secondary. Only models with still more pronounced limb brightening, equivalent to BT90m with m <  0.3, would yield a still larger value of a_irr (and a still smaller primary mass). An estimate of the remaining systematic errors is given in Sect. 7.5.

7.3. Narrow emission lines in HY Eri

In its 2008 low state, HY Eri displayed emission lines of hydrogen, HeI, and Mg Iλ5170. The near infrared Ca II triplet was not detectable against the cyclotron background. In Hα, the broad component with a FWHM of 23 Å and the unresolved narrow component can be separated at our 10 Å spectral resolution, while this becomes infeasible for the higher Balmer lines, which are embedded in complex Zeeman absorption troughs. Mg Iλ5170 is not disturbed and is the only metal line that is sufficiently strong for a radial velocity study. The left panels in Fig. 9 show pseudo-trailed spectra of Hα and Mg Iλ5170 with the phase-dependent continuum subtracted. The gray scale is inverted compared with Fig. 5. We used spectral set 2 with 20 spectra per orbit for Hα and set 1 with 10 spectra per orbit for the weaker Mg I line. The two orbits were folded and the data shown twice for better visibility.

Fig. 9. Left: Hα and Mg Iλ5170 emission lines derived from the spectropolarimetry of 31 Dec 2008, with the neighboring continuum subtracted and shown twice for better visibility. Center: corresponding Doppler maps computed by the maximum entropy method MEM. The overlays show the Roche lobes of the secondary for the best bloated model of Sect. 7.5. Top right: the first orbit shows the orbital flux variations of the Mg I line (blue) and the narrow component of Hα (green). The second orbit shows the Hα flux and its mirror image around ϕ = 0.5 (see text). Bottom right: radial velocity curves of the MgI line (blue), the narrow Hα component (green), and the broad Hα component (crosses). A typical error bar for the broad line is shown in the lower left.

The upper right panel in Fig. 9 shows the orbital flux variations of MgIλ5170 (blue) and of the narrow Hα component (green). Maximum flux occurs near ϕ = 0.5, when the illuminated hemisphere is in view. Half an orbit later, the Mg line disappears and the narrow Hα component can no longer be discriminated against the underlying broad component (yellow). The upper half of the light curve is skewed, as is illustrated in the second orbit, where the Hα light curve is compared with its own mirror image relative to ϕ = 0.5 (dashed curve). The skew is, at least in part, due to statistical fluctuations between the two orbits. Apart from this, the light curve is well described by the irradiation model BR08 (solid curve). The lower right panel of Fig. 9 shows the radial velocity curves of MgIλ5170 (blue), of the narrow component of Hα (green), and of the broad Hα emission (crosses). Although MgIλ5170 is weaker than Hα, the radial velocities have similar errors because the former were derived from single-Gaussian and the latter from the more uncertain double-Gaussian fits. MgIλ5170 has a radial velocity amplitude of $K_2^{\prime }=139\pm 10$ km s⁻¹ with a blue-to-red zero-crossing phase of ϕ₀ = 0.02 ± 0.02, the narrow Hα line has $K_2^{\prime }=125\pm 9$ km s⁻¹ with ϕ₀ = −0.06 ± 0.02. The broad component has a velocity amplitude K_broad = 220 ± 14 km s⁻¹ with ϕ₀ = −0.13 ± 0.02 and γ = −21 ± 9 km s⁻¹ relative to the narrow component. All errors refer to the 90% confidence level.

We investigated the origin of the lines by calculating Doppler tomograms, using the maximum entropy method MEM (Spruit 1998; Marsh & Schwope 2016). Given the small number of phase intervals, the tomograms are sensitive to noise and have been slightly smoothed with a velocity filter corresponding to 0.3 spectral resolution elements. The resulting tomograms are shown in the center panels of Fig. 9, with the outline of the Roche lobe of the secondary for our best-fitting dynamical model overplotted (Table 9, line 4). The bulk of the emission can be uniquely allocated to the illuminated face of the secondary star and the vicinity of the inner Lagrangian point L₁. The rainbow color scale ranges from blue for the highest intensity down to red. Overall, the Hα tomogram is tilted toward the leading hemisphere of the secondary, with the asymmetry related to the finite negative ϕ₀ and the existence of the underlying broad component. The latter with its best visibility at ϕ = 0.60 is represented by the tail that extends to V_X = −700 km s⁻¹ and V_Y = +450 km s⁻¹. This direction differs significantly from that of the standard ballistic stream seen in many polars in their high states, which moves in velocity space from L₁ to large negative V_X at nearly constant V_Y. A tail at similarly odd velocities was seen in the He IIλ4686 tomogram of AM Her (Staude et al. 2004) and tentatively interpreted in terms of a non-standard accretion stream that couples from the secondary immediately to a polar field line of the WD. This is an intriguing proposition in view of our suggestion in Sect. 6.4 that pole 2 of HY Eri is fed by such a scenario. In view of these idiosyncrasies, we should be wary of interpreting the narrow Hα line in HY Eri as of purely chromospheric origin.

The Mg Iλ5170 line is free of the complications by a broad component, the tomogram looks more regular, and the phase of zero radial velocity is consistent with inferior or superior conjunction of the secondary star. The bottom center panel of Fig. 9 shows the enlarged central portion of the tomogram. The emission is centered on the illuminated part of the star, with the peak intensity occurring at a Y-velocity that agrees with $K_2^{\prime }=139\pm 10$ km s⁻¹ obtained from the radial-velocity analysis (small white cross).

7.4. Mass-radius relation of the secondary star

Deriving stellar masses requires that we adopt a mass-radius relation R₂(M₂) for the Roche-lobe filling secondary star. We used theoretical models by Baraffe et al. (1998, 2015, henceforth BCAH and BHAC) for main sequence stars of solar composition evolved to 1 Gyr. This is the approximate cooling age of the WD in HY Eri, discounting compressional heating by accretion, and the minimal age of the secondary. For ease of application, we represented the radii by power laws R₂(M₂)¹¹ Since secondary stars in CVs are known to be more or less bloated compared with field stars, we considered radii expanded over those of the BHAC models by the following processes: (i) magnetic activity and spot coverage (Chabrier et al. 2007; Morales et al. 2010; Knigge et al. 2011; Parsons et al. 2018); (ii) tidal and rotational deformation of Roche-lobe filling stars (Renvoizé et al. 2002); and (iii) inflation by magnetic braking that drives the star out of thermal equilibrium (Knigge et al. 2011). Effect (i) describes the radius excess that compensates for the reduced radiative efficiency caused by starspots. Knigge et al. (2011) and Parsons et al. (2018) found a mean excess of 5% for stars with mass below 0.35 M_⊙. Morales et al. (2010) and Knigge et al. (2011) argued that high-latitude spots may mimic a larger radius in certain eclipsing binaries, accounting for 3% of the excess. Proceeding conservatively, we accept this argument and adopted a bloating factor f₁ = 1.020. A Roche-lobe filling star in a short period binary can not escape effect (ii), which increases the radius by a factor f₂ = 1.045 independent of q (Renvoizé et al. 2002; Knigge et al. 2011). Effect (iii) is described by a free factor f₃ that may range from unity up to about 1.30. The adopted stellar radii R₂ = f₁f₂f₃ R_BHAC are fully consistent with those employed by Knigge et al. (2011) in their evolutionary sequences. Using the models of stars with solar composition evolved to ages of 5 or 10 Gyr instead of 1 Gyr, the dynamical solution yields WD masses lower by 3% or 5%, respectively. For a metal-poor secondary with [M/H] = −1, the masses would be higher by 8% at 1 Gyr, but correspondingly lower again for larger ages.

7.5. Component masses and distance

For a given irradiation model and a mass-radius relation of the secondary star, we obtained the system parameters that match the radial-velocity amplitude $K_2^{\prime }$ and the eclipse duration Δ t_ecl. We adopted $K_2^{\prime }=139\pm 10$ km s⁻¹ of the MgIλ5170 line and BR08 as the standard. Results are presented in Table 9 and Fig. 10. The derived parameters include the masses and radii of the components, the distance d obtained from the angular radius of the secondary star (Sect. 5.2) and the spectroscopic radius R_1, sp of the WD obtained from d and the angular radius of the WD (Sect. 6.5). The listed model radii in Cols. (12) and (13) of Table 9 refer to He-core and CO-core WDs with thick hydrogen envelopes¹² evolved to T_eff = 10 kK (Panei et al. 2007; Renedo et al. 2010; Althaus et al. 2013)¹³. The radii for an effective temperature of 9 kK are only minimally smaller.

Fig. 10. Dynamic models of HY Eri in the M1 − M2 plane. The irradiation model is BR08, $K_2^{\prime }$ ranges from 111 to 159 km s−1 in steps of 2 km s−1, and the bloating factor f3 ranges from 1.000 to 1.300 in steps of 0.005. The models with main sequence (MS) secondaries are marked by blue dots. Red dots denote the models that match both, the measured radial-velocity amplitude of Mg Iλ5170 and the spectroscopically measured WD radius within their 90% confidence errors.

To start with, we take the secondary to be an unbloated main sequence star with f₃ = 1.0, an assumption that yields the maximum primary mass, but disregards the spectroscopic information from Sect. 6.5. For the Mg line with $K_2^{\prime }=139\pm 10$ km s⁻¹ and BR08, the component masses are M₁ = 0.497 ± 0.029 M_⊙ and M₂ = 0.305 ± 0.002 M_⊙ (Table 9, lines 1 to 3, 90% confidence errors). Interestingly, the Hα value $K_2^{\prime }=125\pm 9$ km s⁻¹ and BT90, yield practically the same primary mass, M₁ = 0.487 ± 0.026 M_⊙, but this combination lacks the internal consistency that exists between the Mg I line and the metal-line calibrated model BR08. Cross-combining velocity amplitude and irradiation model gives an indication of the remaining systematic error: Hα and BR08 give M₁ = 0.458 M_⊙, Mg I and BT90 give M₁ = 0.528 M_⊙, or combined M₁ = 0.493 ± 0.035 M_⊙. Hence the so-defined systematic error is of the same size as the statistical error of 0.029 M_⊙. In summary, the assumption of a main sequence secondary identifies the primary either as a He-core WD or a CO-core WD very close to its minimum mass of 0.53 ± 0.02 M_⊙ (Moehler et al. 2004; Kalirai et al. 2008). The fault with the main sequence assumption is the neglect of the spectroscopic evidence of Sect. 6.5 on the angular radius and the effective temperature of the WD. The implied model radius of the primary in either Cols. (12) or (13) of Table 9, lines 1 − 3, falls far short of the spectroscopically determined radius R_1, sp in Col. (15), calculated from R_wd/d = (1.15 ± 0.14)×10⁹ cm kpc⁻¹ (Sect. 6.5) and the R₂-dependent distance d in Col. (14), where we have added the errors quadratically. The employed angular radius of the WD belongs to the best-fit photospheric temperature T_ph = 9.2 ± 0.7 kK. Had the WD the radius of Cols. (12) or (13), the observed spectral flux would demand that its temperature would be that in Col. (18). As noted in Sect. 6.5, a decent Zeeman spectral fit can not be achieved for T_ph >  10 kK, further reducing the probability that the primary in HY Eri is a low-mass CO WD.

In a second step, we considered models with bloated secondary stars. We calculated a grid of models with radial velocity amplitudes $K_2^{\prime }=110$ to 160 km s⁻¹ and expansion factors f₃ = 1.0 to 1.3 in steps of 2 km s⁻¹ and 0.005, respectively. We identified $K_2^{\prime }$ with the radial velocity amplitude of the Mg Iλ5170 line and converted it to K₂, using the BR08 model. The resulting component masses are depicted in Fig. 10, where each model is represented by a dot. The main sequence models considered above are marked in blue. Models that comply with the Mg I amplitude $K_2^{\prime }=139\pm 10$ km s⁻¹, are located between the two dashed lines, extending from the upper right to the lower left. Along this path, the bloating factor f₃ of the secondary star increases, the mass of the WD decreases, its radius increases, and its temperature decreases. Models with He-core WDs, whose radii agree within the uncertainties with the spectroscopically determined WD radius, R_1, He/R_1, sp = 1.00 ± 0.14 (90% confidence error, Sect. 6.5) are located between the two dashed lines that run from the upper left to the lower right, and models that match both conditions are marked by red dots. The optimal dynamical model in line 4 of Table 9 corresponds to the intersection of the two solid lines and the WD parameters at this point correspond to those of the optimal multipole Zeeman fit in Table 8. Lines 5 − 8 of Table 9 contain the model parameters for the four cardinal points of the red-dotted region, marked by the four black dots. Column (3) of the table lists the bloating factor f₃. Bloating ranges from a minimal 3.5% to 19%, indicating that the secondary is only moderately expanded as may be expected for a polar that experiences reduced magnetic braking (Wickramasinghe & Wu 1994; Webbink & Wickramasinghe 2002). As discussed in Sect. 8, minimal bloating that goes along with a low accretion rate is also required to explain the low photospheric temperature of the WD in terms of compressional heating (Townsley & Gänsicke 2009). At the optimal position and at two of the cardinal points, the model and observed WD radii in Cols. (12) and (15) agree. At the two other cardinal points, they disagree by the permitted ±14% (Col. 16). The 90% confidence region for the combined dynamical and spectroscopic fit is defined by a quasi-ellipse that is inscribed to the red-dotted quadrilateral and passes through the cardinal points (not shown). It limits the component masses to M₁ = 0.413  + 0.058, −0.044 M_⊙ and M₂ = 0.235 + 0.048, −0.038 M_⊙ (90% confidence errors), or M₁ = 0.42 ± 0.05 M_⊙ and M₂ = 0.24 ± 0.04 M_⊙, with q between 0.51 and 0.63. The combined dynamical and spectroscopic fit identifies the primary in HY Eri as a low-mass WD, consistent with having a helium core. The mass of the secondary is normal for a CV with an orbital period of 2.855 h. It may be fully convective or retain a radiative core, depending on its prehistory (Knigge et al. 2011).

The observed parameters of HY Eri are summarized in Table 10. At a distance d = 1050 ± 110 pc and a galactic latitude b = −26.1°, it is located close to 500 pc below the galactic plane. The second Gaia data release (Gaia Collaboration 2018) yielded a parallax π = −0.1 ± 0.8 mas at a mean g = 20.3, or a distance of d >  830 pc (Bailer-Jones et al. 2018), consistent with all entries in Table 9.

7.6. Luminosity and accretion rate

The spectral energy distribution (SED) in Fig. 11 provides an overview of the long-term variability of HY Eri. It shows the spectrum of 1993 from Fig. 3 (black curve), that of 2008 from Fig. 7 (red), and the eclipse spectrum from the center panel of Fig. 2 (blue). The photometric data were accessed via the Vizier SED tool provided by the CDS¹⁴. They include the SDSS¹⁵, the UKIDSS, and the Wide-field Infrared Survey (WISE; Cutri et al. 2013) as cyan blue dots, the Two Micron All Sky Survey (2MASS; Skrutskie et al. 2006) as green dots, the Galaxy Evolution Explorer (GALEX; Martin et al. 2005; Bianchi et al. 2011) as blue dots, and the catalogs PPMXL (Roeser et al. 2010) and NOMAD (Zacharias et al. 2005) as red dots. The range of the MONET WL measurements of Fig. 1 is indicated by the four open triangles that refer to the observations of January 2010, November 2010, 2017/18, and the remnant flux inside the eclipse. They delineate the same range of flux levels as the independent photometric and spectrophotometric observations. The secondary star (+) represents a lower flux limit in the red part of the SED.

Fig. 11. Spectral energy distribution of HY Eri from the far ultraviolet to the near infrared based in part on publicly available photometric data accessed via the Vizier SED tool available at the CDS in Strasbourg.

To a first approximation, the high state is characterized by a roughly flat SED from the infrared to the FUV, with F_ν ≃ 0.5 mJy and an integrated energy flux of F_UV, opt ≃ 1.5 × 10⁻¹¹ erg cm⁻² s⁻¹. At a distance of 1 kpc, the high-state UV–optical luminosity amounts to L_UV, opt ≃ 2.0 × 10³³ erg s⁻¹ for emission into 4π. The 2002 UV and visual photometry with the XMM-Newton optical monitor demonstrates that the simultaneous X-ray observation was also taken in a near-high state. The bolometric X-ray flux measured in the ROSAT and XMM-Newton observations is not well established. In Sect. 4 we quoted a conservative high-state level of the bolometric soft X-ray flux of F_X ≃ 1.5 × 10⁻¹¹ erg cm⁻² s⁻¹, although the X-ray flux may have been significantly higher at times. Hence, as a conservative estimate the total high-state luminosity, not accounting for the hard X-ray region, was L_{X, UV, opt} ≃ 4 × 10³³ erg s⁻¹. For the best-fit model in Table 9, line 4 and Table 10 with a WD of 0.42 M_⊙, this luminosity requires an accretion rate of Ṁ ≃ 9 × 10¹⁶ g s⁻¹ = 1.4 × 10⁻⁹ M_⊙ yr⁻¹, On the other hand, the integrated orbital mean cyclotron flux in the 2008 low state, including the extrapolation into the unobserved infrared regime, amounted to F_opt, IR ≃ 3 × 10⁻¹⁴ erg cm⁻² s⁻¹. The implied remnant accretion rate was a mere Ṁ ≃ 1.4 × 10⁻¹² M_⊙ yr⁻¹, three orders of magnitude lower than in the high state.

8. Discussion

HY Eri belongs to the rather small group of about a dozen eclipsing polars that may be considered well studied (Ritter & Kolb 2003, final edition 7.24, 2016). When looked at superficially, it is a rather uninspiring member of the class, being distant and faint, and lapsing into prolonged states of low accretion¹⁶. The present study, however, reveals HY Eri as a system that is peculiar in several respects and in part unique among polars.

We identified it as a permanent two-pole accretor with both poles being active for accretion rates differing by three orders of magnitude. Two nearly opposite accretion spots with similar field strengths seem to suggest a dipolar field, but that is probably an illusion. Our combined quasi-dipole fit to the spectra of both poles is formally good and yields field directions in both accretion spots that are consistent with the results of our cyclotron line fits, but nevertheless the magnetic geometry is characterized by tightly closed field lines in both spots that seem to preclude accretion. In an extensive grid search, we discovered an alternative close-to-perfect fit for a field structure with a polar field strength of the dipole B_dip = 40.4 MG and relative field strengths of quadrupole and octupole B_qua/B_dip = ∓0.14 and B_oct/B_dip = −0.44. It possesses open field lines in both spots, allowing the southern spot 1 to accrete from the nearby orbital plane, while the field in the northern spot 2 leads over the rotational pole to a place somewhere near the secondary star. If that path is active, it obviates the need for the plasma accreted at spot 2 to travel halfway around the WD until it attaches to a near-polar field line. We caution, however, that models that include the full zoo of tesseral harmonics may favor still different magnetic geometries. Furthermore, Zeeman fits based on intensity spectra alone may not yield a unique result. The simultaneous fit to intensity and circular polarization spectra offers better perspectives as demonstrated by Euchner et al. (2002, 2005, 2006) and Beuermann et al. (2007).

The Zeeman spectra of 2008 were obtained in a state of low accretion and yielded a mean mass-flow rate ṁ close to that of the bombardment solution of Woelk & Beuermann (1992). The spot fields obtained from the Zeeman and cyclotron fits differ by about 3%, suggesting that the cyclotron emission originates at a height above the photosphere of roughly 0.01 R_wd, provided both originate at the same position on the WD. This altitude is in the same ballpark as the shock height of ∼0.006 R_wd predicted by two fluid cooling theory (Fischer & Beuermann 2001) for the mass flow rate listed in Table 6.

The property by which HY Eri deviates most drastically from other polars, is the low primary mass of M_wd = 0.42 ± 0.05 M_⊙, based on the simultaneous dynamical and Zeeman spectral analysis. The dynamical analysis alone limits the primary mass to M₁ <  0.53 M_⊙. The secondary mass of M₂ = 0.24 ± 0.04 M_⊙ appears normal for a CV of 2.855 h orbital period. Only two other polars in the Ritter & Kolb catalog (Ritter & Kolb 2003, final edition 7.24, 2016) have reported WD masses below 0.50 M_⊙. Gänsicke et al. (2000) derived the angular radius of the WD in V1043 Cen from far-ultraviolet spectroscopy and the radius and mass from a distance estimate of 200 pc. At the Gaia distance of 172 pc (Gaia Collaboration 2018), however, the radius of the WD is 9.5 × 10⁸ cm and, at a temperature of 15 000 K, the implied mass of 0.57 M_⊙ is consistent with a carbon-oxygen interior. Schwope & Mengel (1997) identified a narrow emission line in EP Dra, which they assigned to the irradiated face of the secondary star. The velocity amplitude $K_2^{\prime }=210\pm 25$ km s⁻¹, led to a primary mass of 0.43 ± 0.07 M_⊙. Since the narrow line was not detected over the entire orbit, an independent confirmation is desirable. Hence, with the possible exception of EP Dra, HY Eri may be the only polar with good evidence of a low-mass primary.

A long-standing discrepancy exists between the large number of CVs with low-mass primary stars predicted by binary population synthesis models and the fact that none has been definitely detected so far (e.g., Zorotovic et al. 2011). Schreiber et al. (2016) employed an empirical version of the concept of consequential angular momentum loss (eCAML) that enhances the standard AML caused by magnetic braking and gravitational radiation. The considered CAML is related to nova outbursts and affects primarily low-mass CVs, removing them preferentially from the population. Nelemans et al. (2016) considered asymmetric nova explosions that could provide a kick to the WD and enhanced the mass transfer rate by the ensuing ellipticity of the orbit. Schenker et al. (1998) considered a Bondi-Hoyle type frictional interaction of the secondary with the nova envelope that transfers orbital angular momentum to the shell and may ultimately drive the system over the stability limit. The AML adopted by Schreiber et al. (2016) varies as C/M₁, with C a free parameter. For appropriate C, it drives low-mass CVs into catastrophic mass transfer and leaves CVs with massive primaries unaffected. The frictional AML experienced by the secondary star moving in the nova shell is proportional to the ejected mass, which varies approximately as $R_1^4/M_1$. It is furthermore proportional to the duration of the interaction, that is, inversely proportional to the expansion velocity V_ex of the envelope. The mean expansion velocity can exceed 1000 km s⁻¹ for high-mass WDs, but is much lower for nova events on low-mass WDs, V_ex ≃ 100 − 200 for 0.6 M_⊙ and “a few tens km s⁻¹ ” for 0.4 M_⊙ (Shara et al. 1993). A frictional AML that rises steeply with decreasing mass is, therefore, a plausible proposition. The model may explain the lack of low-mass CVs provided friction proves sufficiently effective or an alternative mechanism as the one of Nelemans et al. (2016) can be identified.

With a mass ratio q = 0.57 ± 0.06, HY Eri stays just below the stability limit at q ≃ 0.65 (Nelemans et al. 2016; Schreiber et al. 2016), ensuring stable mass transfer in the absence of frictional AML. The added eCAML may render it unstable, if sufficient angular momentum can be extracted from the orbit. The long time scales in low-mass CVs imply, however, that instability is delayed from the time mass transfer started, at least by the waiting time until the first nova outburst and possibly longer. This delay holds similarly for all nova-related CAML descriptions. Nova outbursts in polars with a low-mass primary and typically a rather low accretion rate are especially rare because the ignition mass ΔM_ig increases (i) with decreasing Ṁ and (ii) with decreasing M₁ approximately as $R_1^4/M_1$ (Townsley & Bildsten 2004, their Fig. 8). For nova outbursts on a He-core WD of 0.4 M_⊙, Shara et al. (1993) found Δ M_ig ≃ 9 × 10⁻⁴ M_⊙. Such a large ignition mass implies that nova outbursts in HY Eri have a recurrence time in excess of 10⁷ yr, given the long-term mean accretion rate of HY Eri of 5 × 10⁻¹¹ M_⊙ yr⁻¹ (see below). Hence, HY Eri may have accreted already for 10 million years without having experienced a nova outburst that lured it on to destruction. This time span amounts to ∼5% of the typical 2 × 10⁸ yr it takes non-magnetic CVs to reach the period gap (Knigge et al. 2011). Hence, most CVs with a low-mass primary, must have met their fate much earlier than HY Eri if the eCAML model correctly describes the evolution of CVs. The existence of HY Eri suggests that it is either young and still doomed to destruction or is somehow peculiar, in having managed to escape that outcome.

By its mass, the primary in HY Eri is consistent with a helium WD, but residing in a close binary, it could well be a hybrid star with a mixed He-CO composition (Prada Moroni & Straniero 2009; Zenati et al. 2019). Such WD are created by massive mass loss of the progenitor star that interrupts He-burning after the star has passed the tip of the RGB. The hydrogen-deficient core mass is about 0.46 M_⊙, when stars with an initial mass M_i ≲ 2 M_⊙ incur the He flash, but for initial masses of 2.2 − 2.6 M_⊙, He burning starts at a core mass of 0.32 − 0.34 M_⊙ and subsequent rapid mass loss in a common envelope event could create a WD with the mass of the primary in HY Eri that has a mixed He–CO composition.

With a period of 2.855 h, HY Eri is nominally located in the 2.1−3.1 h period gap (Knigge et al. 2011), but the high-state accretion rate Ṁ ≃ 1.4 × 10⁻⁹ M_⊙ yr⁻¹ (Sect. 7.6) demonstrates that HY Eri has not yet entered its period gap, if it ever will. Wickramasinghe & Wu (1994) and Webbink & Wickramasinghe (2002) argued that polars experience a lower angular momentum loss by magnetic braking than non-magnetic CVs because trapping of the wind from the secondary star in the WD magnetosphere reduces the braking efficiency. They predicted that the gap disappears for primaries with sufficiently high magnetic moments, μ ≳ 4 × 10³⁴ Gcm³. The WD in HY Eri has μ <  3.6 × 10³⁴ Gcm³, based on R_wd ≃ 1.22 × 10⁹ cm and a polar field strength B_p ≃ 40 MG of the dipole component (Table 8). This may or may not suffice to suppress magnetic braking and dispose of the gap. Even if HY Eri were about to enter the gap at a period of 2.85 h, the delayed entry compared with the standard upper edge of the gap of 3.1 h is easily explained by a reduced level of magnetic braking. Given, for example, a braking efficiency reduced by a factor of two over that advocated by Knigge et al. (2011, their Fig. 14, lower panel), the upper edge of the gap would shift down to 2.85 h. A low metalicity of the secondary would add to a delayed entry into the gap (Webbink & Wickramasinghe 2002). If such a scenario applies to HY Eri, the star may still reside above its own gap and await entering it at some time in the future, if at all.

The telling argument against strong magnetic braking is the low photospheric effective temperature of the WD of 9 − 10 kK. The theory of compressional heating relates T_eff to the mean accretion rate ⟨Ṗ⟩ averaged over the Kelvin-Helmholtz time scale of the non-degenerate envelope, which is of the order of 10⁶ yr for a low-mass WD (Townsley & Gänsicke 2009). If the star succeeds in establishing an equilibrium between heating and cooling, its quiescent luminosity is L_q = 6 × 10⁻³ L_⊙ ⟨Ṁ⟩₋₁₀ (M₁/M_⊙)^0.4 (Townsley & Gänsicke 2009, their Eq. (1)), where ⟨Ṁ⟩₋₁₀ is the accretion rate in units of 10⁻¹⁰ M_⊙ yr⁻¹. Provided compressional heating dominates over the congenital heat reservoir, we may equate L_q to $4\pi R_1^2\sigma T_{\text{ph}}^4$. With R₁  =  6.56 × 10⁸(M₁/M_⊙)^−0.60 for M₁  =  0.4 − 0.6 M_⊙ (Althaus et al. 2013), we obtain $T_{\mathrm{ph}}=16.7{⟨\dot{M}⟩}_{-10}^{1/4}{(M_1/M_{\odot })}^{0.40}$ kK. For M₁  =  0.42 M_⊙ (Table 10), we find that ⟨Ṁ⟩ = 5 × 10⁻¹¹ M_⊙ yr⁻¹ suffices to keep the temperature of the WD at 9−10 kK. In its high state, HY Eri reached more than 10⁻⁹ M_⊙ yr⁻¹, implying that the long-term mean duty cycle must be heavily weighted toward states of low accretion. The moderate mass loss of the secondary, the reduced magnetic braking, and the moderate inflation suggested by our best-fit seem to be in line with the evolution of polars as envisaged by Webbink & Wickramasinghe (2002).

HY Eri experienced a highly significant and so far unexplained change of its orbital period by 10.5 ms between 2011 and 2018. Similar period variations have been observed in other post common-envelope binaries (PCEB). Attempts to explain the observations involve either (i) the action of additional bodies encircling the binary, causing an apparent period variation, or (ii) solar-cycle like variations in the internal constitution of the secondary star that change its quadrupole moment and the gravitational pull on the primary, leading to a genuine period variation (Applegate 1992; Völschow et al. 2018; Lanza 2020, and references therein). Process (i) is the likely explanation for at least part of the period variations in NN Ser (Beuermann et al. 2013; Bours et al. 2016), but utterly fails in others like QS Vir (Bours et al. 2016), for which no stable planetary model was found, even considering retrograde and highly inclined orbits (S. Dreizler, priv. comm.). The finding of Bours et al. (2016) that PCEB with convective secondaries of spectral type later than M5.5 largely lack period variations seems to favor magnetic cycles as the driving mechanism. The mechanism of Applegate (1992) and the variants of Völschow et al. (2018) and Lanza (2020) are appealing because they are based on physical processes known to exist in late-type stars, but most authors agree that they are too feeble to produce the observed amplitudes of the period variations in many CVs (Völschow et al. 2018; Lanza 2020). It has also been argued that PCEB may be a natural habitat of circumbinary planets (Völschow et al. 2014). The wealth of period variations observed in PCEB and RSCVN binaries may well have more than a single physical cause. The data presently available for HY Eri are not sufficient to draw definite conclusions on the origin of the observed period variation.

In summary, HY Eri is a rare example of the polar subgroup of magnetic CVs that harbors a low-mass primary, either a helium WD or a hybrid He-CO WD. The system may have passed through the common-envelope phase after severe mass loss on the giant branch or during initial He-burning (Prada Moroni & Straniero 2009; Zenati et al. 2019). The key experiment to prove the low mass of the WD would be a direct measurement of its radius. Unfortunately, our Sloan griz photometry of 2017 lacked the time resolution to measure the radius of the WD from the finite ingress and egress times of the eclipse light curves in g or possibly r. With HY Eri still in a prolonged state of low accretion as of early 2019, this task could be accomplished by high-speed photometry at a large telescope. If the low mass of the WD is confirmed, a dedicated evolutionary study could establish the origin and future evolution of HY Eri.

Acknowledgments

We dedicate this paper to the lasting memory of Hans-Christoph Thomas, who analyzed part of the early data described here before his untimely death on 18 January 2012. We thank the anonymous referee for the careful reading of the paper and valuable comments that helped to improve this work. Part of the data were collected with the telescopes of the MOnitoring NEtwork of Telescopes, funded by the Alfried Krupp von Bohlen und Halbach Foundation, Essen, and operated by the Georg-August-Universität Göttingen, the McDonald Observatory of the University of Texas at Austin, and the South African Astronomical Observatory. We made use of the ROSAT Data Archive of the Max-Planck-Institut für Extraterrestrische Physik (MPE) at Garching, Germany. Part of the analysis is also based on observations of RE J0501-03 obtained with XMM-Newton on 2002-03-24, Obs-Id 0109460601. The observations at ESO in the years 2000, 2001, 2008, 2016, 2017, and 2018 were collected at the La Silla and Paranal sites under the programme IDs 66.D-0128, 66.D-0513, 082.D-0695, 098.A-9099, and 0100.A-9099. In establishing the spectral energy distribution in Fig. 11, we accessed various data archives via the VizieR Photometric viewer operated at the CDS, Strasbourg, France (http://vizier.u-strasbg.fr/vizier/sed/). We quoted distances for HY Eri and EP Dra from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia).

References

All Tables

All Figures

	Fig. 1. Left, from top: orbital light curves of HY Eri in the 1996 and January 2010 high states, November 2010 intermediate, and 2017 low state, binned to ∼2 min time resolution. Right, from top: binned ROSAT X-ray light curves measured with either the PSPC or the HRI as detectors and XMM-Newton light curve measured with the MOS and pn detectors of the EPIC camera.
In the text

	Fig. 2. Left: eclipse light curves in the high, intermediate, and in the low state, the former two shifted upward to avoid overlap. Phases are from Eq. (1) for the high and intermediate states and from and Eq. (2) for the low state. Right: O − C diagram for the deviations of the mid-eclipse times from the linear ephemeris of Eq. (1), showing the change in orbital period.
In the text

Fig. 3. Top: mean flux-calibrated low-resolution spectra of HY Eri in the high states of 1993 and 1995. For comparison, the eclipse spectrum of 2000 (blue curve) and the mean out-of-eclipse spectrum in the 2008 low state (red curve) are added on the same ordinate scale. Bottom: mean radial velocities of the broad emission lines of Hβ and HeIIλ4686 (open circles) and of the line peaks (cyan dots) derived from medium-resolution spectra taken on 15 November 1995.

In the text

Fig. 4. Top: eclipse light curve of the Hα emission line flux in erg cm−2 s−1 and of the underlying continuum flux observed on 20 November 2000. Center: eclipse spectrum (black curve) thermal hydrogen spectrum adjusted to fit the Hα line flux (blue or red curve, see text). The ordinate is in units of 10−16 erg cm−2 s−1 Å−1. Bottom: difference spectra on the same color coding fitted by a dM6 star (red) and a dM5 star (blue, shifted upward by 0.05 units).

In the text

Fig. 5. Left: intensity spectra of the 2008 spectropolarimetry with the contribution of the secondary star subtracted. The mean spectra of two orbits are shown twice for visual continuity. Two systems of cyclotron lines are visible that originate near two poles of similar field strength. Center: circular polarization spectra with white indicating positive and black negative polarization. Right: maxima of the cyclotron lines of both poles (circles, triangles), extrema of the circular polarization (plus signs, crosses), and best-fit line positions as functions of orbital phase (red curves).

In the text

Fig. 6. Left: mean bright-phase dereddened intensity spectrum of the 2008 spectropolarimetry with the contribution of the secondary star subtracted (red curve). For comparison, the appropriate model spectrum of a magnetic WD is shown (black curve). The mean fluxes of the November 2017 and February 2018 gr photometry are included as the two open blue circles. The red numbers indicate the cyclotron harmonics. Right: light curves for selected wavelength intervals derived from spectral set 1 before phasefolding, but after subtraction of the secondary star. The individual panels show the quasi-B band flux, the cyclotron beaming of the pole-1 and pole-2 emissions, and the Hα line flux (in arbitrary units).

In the text

Fig. 7. Left: cyclotron spectra of poles 1 and 2 (red curves) least-squares fitted with constant-temperature models for the plasma parameters listed in Table 6 (black curves). Right panels: magnetic geometry of the WD for case A and case B accretion (see text). The secondary star is located to the left. Angles are measured between the dots on the circumference. Dipolar field lines are included as red curves.

In the text

Fig. 8. Left panels: magnetic-field distributions of the WD for the case B multipole model of Table 8 and the phases of best visibility of spots 1S and 2N. The footpoints of the rotational and the magnetic axes are indicated by the black dot and the cross (×), respectively. The magnetic equator and the orbital plane are marked by the dashed and the solid black line, respectively. The individual projected-area elements are marked by tiny dots, selecting or suppressing every second integer field strength. The spot field is marked in green and the viewing-angle selected spot is emphasized in black. The bottom panels show the model field strength vs. the magnetic colatitude ϑ. The red dots mark the spot fields. Right panel: observed pole-1 spectrum (red curve) with the best-fit Zeeman spectrum for the adopted multipole model (black curves). The residuals of the fit are displayed at the bottom with the wavelength sections that are included in the fit highlighted in blue.

In the text

Fig. 9. Left: Hα and Mg Iλ5170 emission lines derived from the spectropolarimetry of 31 Dec 2008, with the neighboring continuum subtracted and shown twice for better visibility. Center: corresponding Doppler maps computed by the maximum entropy method MEM. The overlays show the Roche lobes of the secondary for the best bloated model of Sect. 7.5. Top right: the first orbit shows the orbital flux variations of the Mg I line (blue) and the narrow component of Hα (green). The second orbit shows the Hα flux and its mirror image around ϕ = 0.5 (see text). Bottom right: radial velocity curves of the MgI line (blue), the narrow Hα component (green), and the broad Hα component (crosses). A typical error bar for the broad line is shown in the lower left.

In the text

Fig. 10. Dynamic models of HY Eri in the M1 − M2 plane. The irradiation model is BR08, $K_2^{\prime }$ ranges from 111 to 159 km s−1 in steps of 2 km s−1, and the bloating factor f3 ranges from 1.000 to 1.300 in steps of 0.005. The models with main sequence (MS) secondaries are marked by blue dots. Red dots denote the models that match both, the measured radial-velocity amplitude of Mg Iλ5170 and the spectroscopically measured WD radius within their 90% confidence errors.

In the text

	Fig. 11. Spectral energy distribution of HY Eri from the far ultraviolet to the near infrared based in part on publicly available photometric data accessed via the Vizier SED tool available at the CDS in Strasbourg.
In the text