tokens: toward values of the order of τ c / τ IC ≈ 10 −3 –10 −2 for the relevant electron energies (see e.g., Pereira et al. 2016 ; del Palacio et al. 2018 for a modeling of AE Aur and other bow-shock systems). In particular, Pereira et al. ( 2016 ) claim that for the AE Aur system the energy density of the ambient radiation is largely dominated by dust emission, and found a value for the ratio τ c / τ IC roughly equal to 10 −2 . Finally, values of E max of the order of ≲1 TeV have been estimated in Pereira et al. ( 2016 ), implying that the logarithm in Equation ( 4 ) should be roughly of the order of ≈10. Given these fiducial values we can now estimate the IC flux in the X-ray band aswhich is indeed an upper limit because it is based on the most optimistic value for η e . Our estimate of the IC X-ray emission is consistent with the upper limit reported in Section 3.3 , but is in disagreement with previous and more optimistic estimates reported in the literature (e.g., López-Santiago et al. 2012 ; Pereira et al. 2016 ; del Palacio et al. 2018 ). The main reason for that is the different choice of the parameter η e , which in these previous works was assumed to be equal to η e ≈ 0.1, which implicitly implies that wind termination shocks were assumed to be much more effective than SNR shocks in accelerating electrons. In this respect, our estimate is thus more conservative: we assumed that wind termination shocks behave as SNR ones. To conclude, we stress that a similar reasoning can be adopted (referring to AE Aur) to interpret the radio upper limits. For the fiducial value of the magnetic field of ∼30 μ G, electrons of energy ≳1 GeV emit synchrotron radiation in the GHz domain. The characteristic synchrotron cooling time τ s for such electrons is of the order of several megayears. An estimate of the expected radio emission from the bow shock can then be obtained by substituting τ IC with τ s in Equation ( 5 ), and by noticing that , which is about two orders of magnitude smaller than τ c / τ IC . However, this is almost exactly compensated by the fact that the radio upper limits are much more stringent than those derived from X-ray observations. Therefore, we can conclude that also the nondetection of the bow shock of AE Aur in radio waves is consistent with the assumption that such shocks behave as SNR ones. Within this framework, the detection of radio synchrotron emission from the bow shock of the runaway star BD+43°3654 might be tentatively explained by the very large kinetic energy characterizing its stellar wind. For this system, estimates of the stellar mass-loss rate range from yr −1 (Peri et al. 2012 ) up to 1.6 × 10 −4 M ⊙ yr −1 (Kobulnicky et al. 2010 ), i.e., a factor of ≈30...800 larger than that estimated for AE Aur. Also, the estimated velocity of the wind of BD+43°3654 (2300 km s −1 ) is roughly a factor of two larger than that estimated fro AE Aur (Benaglia et al. 2010 ). Therefore, the wind kinetic energy is ≈10 2 ... 3 × 10 3
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