Diving into the magnetosphere of the Of?p star HD108

We analyse optical and X-ray spectroscopy of the Of?p star HD108, known for
its strong dipolar magnetic field and its optical line profile variability with
a timescale of $54 \pm 3$ yrs, interpreted as the stellar rotation period.
Optical emission lines have now recovered from their minimum emission state
reached in 2007 - 2008. The variations of the equivalent width of the H$\alpha$
emission provide constraints on the inclination of the rotation axis ($i$) and
the obliquity of the magnetic axis ($\beta$). The best agreement between model
and observations is found for ($i$, $\beta$) pairs with $i + \beta \simeq
85^{\circ}$ and $i \in [30^{\circ},55^{\circ}]$. The Balmer emission lines
display stochastic variability at the $\sim 5$ % level on timescales of a few
days. TESS photometry unveils transient modulations on similar timescales in
addition to prominent red noise variations. A Chandra X-ray observation of
December 2021, when the star was at a higher emission level, indicates a slight
increase of the flux and a spectral hardening compared to the August 2002
XMM-Newton observation, taken near minimum emission state. Magnetohydrodynamic
simulations are used to compute synthetic X-ray spectra. With our current best
estimate of the $\dot{M}_{B=0}$ mass-loss rate, the simulated X-ray luminosity
and spectral energy distribution agree very well with the observations.
Finally, the radial velocities vary on a period of 8.5 years with a
peak-to-peak amplitude of 10 - 11 km s$^{-1}$, suggesting orbital motion with
an unseen companion of at least 4 M$_{\odot}$.