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Mass Determination and Detection of the Onset of Chromospheric Activity for the Sub-Stellar Object in EF Eridani

EF Eri is a magnetic cataclysmic variable that has been in a low accretion state for the past nine years. Low state optical spectra reveal the underlying Zeeman-split white dwarf absorption lines. These features are used to determine a value of 13-14 MG as the white dwarf field strength. Recently, 5-7 years into the low state, Balmer and other emission lines have appeared in the optical. An analysis of the H$α$ emission line yields the first radial velocity solution for EF Eri, leading to a spectroscopic ephemeris for the binary and, using the best available white dwarf mass of 0.6M${\odot}$, a mass estimate for the secondary of 0.055M${\odot}$. For a white dwarf mass of 0.95M${\odot}$, the average for magnetic white dwarfs, the secondary mass increases to 0.087M${\odot}$. At EF Eri's orbital period of 81 minutes, this higher mass secondary could not be a normal star and still fit within the Roche lobe. The source of the Balmer and other emission lines is confirmed to be from the sub-stellar secondary and we argue that it is due to stellar activity. We compare EF Eri's emission line spectrum and activity behavior to that recently observed in AM Her and VV Pup and attributed to stellar activity. We explore observations and models originally developed for V471 Tau, for the RS CVn binaries, and for extra-solar planets. We conclude that irradiation of the secondary in EF Eri and similar systems is unlikely and, in polars, the magnetic field interaction between the two stars (with a possible tidal component) is a probable mechanism which would concentrate chromospheric activity on the secondary near the sub-stellar point of the white dwarf.