  The bow shocks of runaway stars with strong stellar winds of over 2000 km
s$^{-1}$ can serve as particle acceleration sites. The conversion from stellar
wind luminosity into particle acceleration power has an efficiency of the same
order of magnitude as those in supernova remnants, based on the radio emission
from the bow shock region of runaway star BD+43$^\circ$3654
\citep{Benaglia10}.If this object exhibits typical characteristics, then
runaway star systems can contribute a non-negligible fraction of Galactic
cosmic-ray electrons. To constrain the maximum energy of accelerated particles
from measurements of possible non-thermal emissions in the X-ray band, Suzaku
observed BD+43$^\circ$3654 in April 2011 with an exposure of 99 ks. Because the
onboard instruments have a stable and low background level, Suzaku detected a
possible enhancement over the background of $7.6\pm 3.4$ cnt arcmin$^{-2}$ at
the bow shock region, where the error represents the 3 sigma statistics only.
However, the excess is not significant within the systematic errors of
non-X-ray and cosmic-ray backgrounds of the X-ray Imaging Spectrometer, which
are $\pm 6.0$ and $\pm 34$ cnt arcmin$^{-2}$, respectively, and the 3-sigma
upper limit in the X-ray luminosity from the shock region, which is $1.1 \times
10^{32}$ erg s$^{-1}$ per 41.2 arcmin$^2$ in the 0.5 to 10 keV band. This
result leads to three conclusions: (1) a shock-heating process is inefficient
on this system; (2) the maximum energy of electrons does not exceed $\sim$ 10
TeV, corresponding to a Lorentz factor of less than $10^7$; and (3) the
magnetic field in the shock acceleration site might not be as turbulent as
those in pulsar wind nebulae and supernova remnants.
