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 nonthermal energy distributions that produce broadband synchrotron emission. At late times, even initially collimated relativistic jets evolve into quasi-spherical Newtonian blast waves. As the shock decelerates, the post-shock internal energy per particle decreases; below a critical velocity β DN ≈ 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 DN ≈ 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 ej = 0.1 M ⊙ ), the DN phase begins prior to deceleration; neglecting it underestimates radio flux by factors of ∼3–5 during coasting and even more thereafter. Magnetar-boosted remnants ( E ∼ 10 52 erg) should reach ∼ 10 – 100 μ Jy at 3 GHz at ∼ 40Mpc, though limits on GW170817 already disfavor a long-lived millisecond magnetar. In core-collapse supernovae in a wind medium ( ρ ∝ r − k ), the peak luminosity remains constant during coasting, while ν pk ∝ t −1 ; for SN 2023ixf, we find k = 1.29 ± 0.14. The DN spectral energy distribution typically satisfies ν m < ν sa < ν c , peaking at sub-GHz frequencies where LOFAR and SKA-low are most sensitive. Even nondetections place robust constraints on ambient density and outflow energetics.