KOI-3890: A high mass-ratio asteroseismic red-giant$+$M-dwarf eclipsing binary undergoing heartbeat tidal interactions

KOI-3890 is a highly eccentric, 153-day period eclipsing, single-lined spectroscopic binary system containing a red-giant star showing solar-like oscillations alongside tidal interactions. The combination of transit photometry, radial velocity observations, and asteroseismology have enabled the detailed characterisation of both the red-giant primary and the M-dwarf companion, along with the tidal interaction and the geometry of the system. The stellar parameters of the red-giant primary are determined through the use of asteroseismology and grid-based modelling to give a mass and radius of M_*=1.04±0.06M_⊙ and R_*=5.8±0.2R_⊙ respectively. When combined with transit photometry the M-dwarf companion is found to have a mass and radius of M_C=0.23±0.01M_⊙ and R_C=0.256±0.007R_⊙. Moreover, through asteroseismology we constrain the age of the system through the red-giant primary to be 9.1^+2.4_-1.7 Gyr. This provides a constraint on the age of the M-dwarf secondary, which is difficult to do for other M-dwarf binary systems. In addition, the asteroseismic analysis yields an estimate of the inclination angle of the rotation axis of the red-giant star of i=87.6^+2.4_-1.2 degrees. The obliquity of the system - the angle between the stellar rotation axis and the angle normal to the orbital plane - is also derived to give ψ=4.2^+2.1_-4.2 degrees showing that the system is consistent with alignment. We observe no radius inflation in the M-dwarf companion when compared to current low-mass stellar models.