tokens: in the downstream. In PIC simulations of nonrelativistic shocks, Bohdan et al. ( 2020 ) found similar values of while Park et al. ( 2015 ) found even more effective thermalization with . This heating via plasma processes of the thermal electrons requires to be large. In Figure 6 , we show as a dashed vertical cyan line, representing the value found by Crumley et al. ( 2019 ) (). This value of would overshoot that radio data by orders of magnitude. Observations of supernova remnants also find the ratio to be between 0.03 and 1, albeit the velocities in those cases are much lower (Ghavamian et al. 2013 ; Vink et al. 2015 ). Very small values of help alleviate the constraint on . However, small values of are problematic in the sense of UHECR acceleration. The dotted–dashed horizontal lines shown in Figure 6 correspond to minimum values of required for UHECR acceleration. They are shown for different values of the best-fit rigidities, similarly to Figure 3 . The lower limit on is calculated using aswhere is the internal energy of the downstream. The lines are plotted for completely stripped iron using acceleration efficiency (thin) and (thick). It is possible that the magnetization decreases with distance to the shock front (e.g., Lemoine et al. 2013 ), something which is not considered in the present work. Inspection of Figure 6 shows that, while , the radio flux is only weakly dependent on . Therefore, for these values of , our conclusion would not change if a varying magnetization was included. As previously mentioned, is suggested by PIC simulations (Sironi Spitkovsky 2011 ; Park et al. 2015 ; Crumley et al. 2019 ; Bohdan et al. 2020 ). The constraints shown in Figure 6 are valid for the microphysical parameters at the forward shock. In principle, the reverse shock could be different from the forward shock, which means that could be sufficiently small in the forward shock as to not overshine the radio, while large enough at the reverse shock for UHECR acceleration to be possible. As both the reverse and the forward shocks are transrelativistic, this might imply that , but this also depends on the magnetization around the two shocks, which can be different. One can estimate as follows. Assuming cosmic ray acceleration occurs during the first shell crossing of the shock, the shell crossing radius is estimated as , where is the shell width and is the central engine duration that we set equal to T 90 (Nakar Piran 2004 ). With an initial Lorentz factor of , the ratio of the ejecta density to the circumburst density is . Thus, the Lorentz factor of the downstream of the reverse shock as seen from a laboratory observer can be estimated as , where has been used (Panaitescu Kumar 2004a ; Zhang Murase 2019 ). 14 14 This approximation for is not very accurate in this case, but it has been used to show the parameter dependence. A more precise value is given by . The magnetic field is calculated as . The value of sufficient to accelerate UHECR can be
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