Neglected X-ray discovered polars: II: The peculiar eclipsing binary HY Eridani

We report on the X-ray observations of the eclipsing polar HY Eri (RX J0501-0359), along with its photometric, spectrophotometric, and spectropolarimetric optical variations, collected over 30 years. With an orbital period of 2.855 h, HY Eri falls near the upper edge of the 2-3 h period gap. After 2011, the system went into a prolonged low state, continuing to accrete at a low level. We present an accurate alias-free long-term orbital ephemeris and report a highly significant period change by 10 ms that took place over the time interval from 2011 to 2018. We acquired a high-quality eclipse spectrum that shows the secondary star as a dM5-6 dwarf at a distance $d = 1050 \pm 110$ pc. Based on phase-resolved cyclotron and Zeeman spectroscopy, we identify the white dwarf (WD) in HY Eri as a two-pole accretor with nearly opposite accretion spots of 28 and 30 MG. The Zeeman analysis of the low state spectrum reveals a complex magnetic field structure, which we fit by a multipole model. We detected narrow emission lines from the irradiated face of the secondary star, of which Mg I $\lambda 5170$ with a radial velocity amplitude of $K_2' = 139 \pm 10$ km/s (90% confidence) tracks the secondary more reliably than the narrow H$\alpha$ line. Based on the combined dynamical analysis and spectroscopic measurement of the angular radius of the WD, we obtain a primary mass of $M_1 = 0.42 \pm 0.05$ $M_\odot$ (90% confidence errors), identifying it as a probable He WD or hybrid HeCO WD. The secondary is a main sequence star of $M_2 = 0.24 \pm 0.04$ $M_\odot$ that seems to be slightly inflated. The large distance of HY Eri and the lack of similar systems suggest a very low space density of polars with low-mass primary. According to current theory, these systems are destroyed by induced runaway mass transfer, suggesting that HY Eri may be doomed to destruction.