  The persistent Be/X-ray binary RX J0440.9+4431 flared in 2010 and 2011 and
has been followed by various X-ray facilities Swift, RXTE, XMM-Newton, and
INTEGRAL. We studied the source timing and spectral properties as a function of
its X-ray luminosity to investigate the transition from normal to flaring
activity and the dynamical properties of the system. We have determined the
orbital period from the long-term Swift/BAT light curve, but our determinations
of the spin period are not precise enough to constrain any orbital solution.
The source spectrum can always be described by a bulk-motion Comptonization
model of black body seed photons attenuated by a moderate photoelectric
absorption. At the highest luminosity, we measured a curvature of the spectrum,
which we attribute to a significant contribution of the radiation pressure in
the accretion process. This allows us to estimate that the transition from a
bulk-motion-dominated flow to a radiatively dominated one happens at a
luminosity of ~2e36 erg/s. The luminosity dependency of the size of the black
body emission region is found to be $r_{BB} \propto L_X^{0.39\pm0.02}$. This
suggests that either matter accreting onto the neutron star hosted in RX
J0440.9+4431 penetrates through closed magnetic field lines at the border of
the compact object magnetosphere or that the structure of the neutron star
magnetic field is more complicated than a simple dipole close to the surface
