2025ApJ...988...30H

Recent observations indicate that stripped-envelope core-collapse supernovae are often surrounded by dense circumstellar material (CSM). Motivated by this, we develop an analytic model to systematically study the dynamics of long gamma-ray burst (LGRB) jet propagation in various CSM environments. We derive a general expression for the jet head velocity (β_h) and breakout time (t_b) valid across Newtonian, relativistic, and intermediate regimes, accounting for a previously unrecognized dependence on 1 - β_h. Our results highlight a fundamental distinction between jet propagation in massive stars, where β_h ≪ 1, and in extended CSM, where 1 - β_h ≪ 1. We establish an analytic success/failure criterion for jets and express it in terms of jet and CSM parameters, revealing a strong dependence on CSM radius. To quantify the relativistic nature of the jet-cocoon system, we introduce the energy-weighted proper velocity $\overline{{\rm{\Gamma }}\beta }$. We identify three possible jet outcomes-(a) successful jets ($\bar{{\rm{\Gamma }}\beta }\sim 10-100$), (b) barely failed jets ($\overline{{\rm{\Gamma }}\beta }\sim 1$), and (c) completely failed jets ($\overline{{\rm{\Gamma }}\beta }\sim 0.1$)-and constrain their respective jet/CSM parameter spaces. We show that, in (b) and (c), large CSM radii can result in luminous fast blue optical transients via cocoon cooling emission. This theoretical framework provides a basis for future observational and theoretical studies to understand the link between LGRBs, intermediate GRBs, low-luminosity LGRBs, and their environments.