Configuration of the $\xi$ Tau system constrained by multi-technique observations

$\xi$ Tau is one of the most compact multiple stellar systems, which is sufficiently close (67 pc) to be constrained by all kinds of observations. To better constrain its current configuration, we utilized new observational data: (i) photometry from TESS and astrometry from WDS, and (ii) our own photometry from the MOST spacecraft and spectroscopy from the CTIO observatory. [...] Given the hierarchical architecture of $\xi$ Tau, ((Aa+Ab)+B)+C, we detected the orbital evolution on all time scales. Oscillations of periods $P$ occur on the shortest, orbital time scales ($P_1$, $P_2$); the variation of eccentricity $e_1$ is from 0 to 0.008, and of $e_2$ from 0.202 to 0.207, respectively. Oscillations of projected $i$, $\Omega$ are coupled, and occur on the secular time scale of about 7000 d. The inclination $i_1$ of the inner, eclipsing pair (Aa+Ab) changes from $86.1^\circ$ to $87.1^\circ$, which is clearly manifested in eclipse depths. There is also a long-term trend due to the outer orbit ($P_3 \doteq 18900\,{\rm d}$), with a perihelion passage (a `bump') of component C, which is manifested in radial velocities. The mutual inclinations between the three orbital planes, ${\simeq}\,0.5^\circ$ and $71^\circ$, are very different. Long-term stability is ensured by suppressing Kozai oscillations due to the fast precession rate $\dot\omega_2$. The best model requires tidal dissipation in the inner binary (with the time lag of ${\sim}100\,{\rm s}$) and five components, where component C is a binary (Ca+Cb). Although the masses of the three components ($2.27$, $2.15$, $3.78\,M_\odot$) are now constrained to within 1%, the suspected binary (Ca+Cb), offset by $600\,{\rm mas}$, should be better characterized. A key question remains whether this bright stellar system contains additional dwarf or exoplanetary components with low masses. Continuing monitoring of $\xi$ Tau is highly desirable.