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 $7.6pm 3.4$ 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.
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