The Deep Newtonian Regime in Late-Time Blast Waves: Inevitable Transition and Distinct Flux Signatures

In many astrophysical transients, outflows drive shocks into the ambient medium, accelerating electrons to non-thermal energy distributions that produce broadband synchrotron emission. At late times, even initially collimated relativistic jets evolve into quasi-spherical Newtonian blastwaves. As the shock decelerates, the post-shock internal energy per particle decreases; below a critical velocity $β_{\rm DN} \approx 0.2$, only a fraction $ξ_e < 1$ of electrons are accelerated to relativistic energies, defining the deep Newtonian (DN) regime. We develop a unified analytic framework for synchrotron emission in this phase, applicable to both single-velocity and stratified ejecta. For gamma-ray burst afterglows in a uniform medium, the DN transition occurs at $t_{\rm DN} \approx 3.7\,E_{51}^{1/3} n_0^{-1/3}$~yr, yielding a shallower decay by $δα= 6(p-2)/5$ relative to standard Newtonian predictions. For kilonova remnants ($E_0 = 10^{50.5}$~erg, $M_{\rm ej} = 0.1\,M_\odot$), the DN phase begins prior to deceleration; neglecting it underestimates radio flux by factors of $\sim 3$--$5$ during coasting and even more thereafter. Magnetar-boosted remnants ($E \sim10^{52}$~erg) should reach $\sim$\,10\,--\,100\,$μ$Jy at 3~GHz at $\sim$\,40\;Mpc, though limits on GW170817 already disfavor a long-lived millisecond magnetar. In core-collapse supernovae in a wind medium ($ρ\!\propto\!r^{-k}$), the peak luminosity remains constant during coasting, while $ν_{\rm pk} \propto t^{-1}$; for SN~2023ixf, we find $k = 1.29 \pm 0.14$. The DN spectral energy distribution typically satisfies $ν_m\!