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 DN about 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 about 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 about 3 -- 5 during coasting and even more thereafter. Magnetar-boosted remnants (E about10 52 erg) should reach about,10,--,100, μ Jy at 3 GHz at about,40;Mpc, though limits on GW170817 already disfavor a long-lived millisecond magnetar. In core-collapse supernovae in a wind medium (ρ! proportional to!r-k), the peak luminosity remains constant during coasting, while ν pk proportional to t-1; for SN 2023ixf, we find k = 1.29 plus or minus 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 non-detections place robust constraints on ambient density and outflow energetics.